{
"query": {
"display": "$$y^{^{\\prime\\prime}}-y^{^{\\prime}}+121y=11\\sin\\left(11t\\right)$$",
"symbolab_question": "ODE#y^{\\prime \\prime }-y^{\\prime }+121y=11\\sin(11t)"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "ODE",
"subTopic": "ConstCoeffLinearNonHomogeneous",
"default": "y=e^{\\frac{t}{2}}(c_{1}\\cos(\\frac{\\sqrt{483}t}{2})+c_{2}\\sin(\\frac{\\sqrt{483}t}{2}))+\\cos(11t)",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right):{\\quad}y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)+\\cos\\left(11t\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right)",
"steps": [
{
"type": "interim",
"title": "Solve linear ODE:$${\\quad}y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)+\\cos\\left(11t\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right)",
"steps": [
{
"type": "definition",
"title": "Second order linear non-homogeneous differential equation with constant coefficients",
"text": "A second order linear, non-homogeneous ODE has the form of $$ay''+by'+cy=g\\left(x\\right)$$"
},
{
"type": "step",
"primary": "The general solution to $$a\\left(x\\right)y''+b\\left(x\\right)y'+c\\left(x\\right)y=g\\left(x\\right)$$ can be written as<br/>$$y=y_h+y_p$$<br/>$$y_h$$ is the solution to the homogeneous ODE $$a\\left(x\\right)y''+b\\left(x\\right)y'+c\\left(x\\right)y=0$$<br/>$$y_p$$, the particular solution, is any function that satisfies the non-homogeneous equation "
},
{
"type": "interim",
"title": "Find $$y_h$$ by solving $$y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=0:{\\quad}y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=0",
"steps": [
{
"type": "definition",
"title": "Second order linear homogeneous differential equation with constant coefficients",
"text": "A second order linear, homogeneous ODE has the form of $$ay''+by'+cy=0$$"
},
{
"type": "step",
"primary": "For an equation $$ay''+by'+cy=0$$, assume a solution of the form $$e^{γt}$$",
"secondary": [
"Rewrite the equation with $$y=e^{γt}$$"
],
"result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}-\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime}}+121e^{γt}=0"
},
{
"type": "interim",
"title": "Simplify $$\\left(\\left(e^{γt}\\right)\\right)^{\\prime\\prime}-\\left(\\left(e^{γt}\\right)\\right)^{\\prime}+121e^{γt}=0:{\\quad}e^{γt}\\left(γ^{2}-γ+121\\right)=0$$",
"steps": [
{
"type": "step",
"result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}-\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime}}+121e^{γt}=0"
},
{
"type": "interim",
"title": "$$\\left(e^{γt}\\right)^{\\prime\\prime}=γ^{2}e^{γt}$$",
"input": "\\left(e^{γt}\\right)^{\\prime\\prime}",
"steps": [
{
"type": "interim",
"title": "$$\\left(e^{γt}\\right)^{\\prime}=e^{γt}γ$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{γt}\\left(γt\\right)^{\\prime}$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"result": "=e^{γt}\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γt$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γt\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γt\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=γt$$",
"result": "=e^{γt}\\left(γt\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcDDfSBXef41nZ6zbP7ViaAosjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5PvoGisVaN+BwjjtpPeRZCLDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "interim",
"title": "$$\\left(γt\\right)^{\\prime}=γ$$",
"input": "\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γt^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7stv4XdlIkdSrTZc5soZLGCENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tSbX/i/cqXdrp84USJNBCUvvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=e^{γt}γ"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=\\left(e^{γt}γ\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(e^{γt}γ\\right)^{\\prime}=γ^{2}e^{γt}$$",
"input": "\\left(e^{γt}γ\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γ\\left(e^{γt}\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{γt}\\left(γt\\right)^{\\prime}$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"result": "=e^{γt}\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γt$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γt\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γt\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=γt$$",
"result": "=e^{γt}\\left(γt\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcDDfSBXef41nZ6zbP7ViaAosjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5PvoGisVaN+BwjjtpPeRZCLDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "interim",
"title": "$$\\left(γt\\right)^{\\prime}=γ$$",
"input": "\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γt^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7stv4XdlIkdSrTZc5soZLGCENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tSbX/i/cqXdrp84USJNBCUvvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=γe^{γt}γ"
},
{
"type": "interim",
"title": "Simplify $$γe^{γt}γ:{\\quad}γ^{2}e^{γt}$$",
"input": "γe^{γt}γ",
"result": "=γ^{2}e^{γt}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$γγ=\\:γ^{1+1}$$"
],
"result": "=e^{γt}γ^{1+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$1+1=2$$",
"result": "=e^{γt}γ^{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7riofKsRQ/FkpZv0BW2pxMd6GQqufR6tr2vPxOUv7H+/BWItNlNCsjK5QfFqGTa8umx4rCXhbsN+br+uOYP22UU3kCh3oevUunZ7/b0qFKBStCRMtul5SOs/SBwPTbaWuo4bl40YraHWFXpFVaYGPXg=="
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=γ^{2}e^{γt}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "γ^{2}e^{γt}-\\left(e^{γt}\\right)^{^{\\prime}}+121e^{γt}=0"
},
{
"type": "interim",
"title": "$$\\left(e^{γt}\\right)^{\\prime}=e^{γt}γ$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}e^{γt}\\left(γt\\right)^{\\prime}$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"result": "=e^{γt}\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γt$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γt\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γt\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=γt$$",
"result": "=e^{γt}\\left(γt\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcDDfSBXef41nZ6zbP7ViaAosjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5PvoGisVaN+BwjjtpPeRZCLDrbw8lc2jRiiaaodUFzB+wS4M5VpC8qh+oehjmM1qmzPHVJGaR3CuIp5NX3rLDDQialcV/dI5TH4fXyp+ncwuA=="
}
},
{
"type": "interim",
"title": "$$\\left(γt\\right)^{\\prime}=γ$$",
"input": "\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γt^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7stv4XdlIkdSrTZc5soZLGCENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tSbX/i/cqXdrp84USJNBCUvvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=e^{γt}γ"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "γ^{2}e^{γt}-e^{γt}γ+121e^{γt}=0"
},
{
"type": "step",
"primary": "Factor $$e^{γt}$$",
"result": "e^{γt}\\left(γ^{2}-γ+121\\right)=0"
}
],
"meta": {
"interimType": "Generic Simplify Specific 1Eq"
}
},
{
"type": "step",
"result": "e^{γt}\\left(γ^{2}-γ+121\\right)=0"
},
{
"type": "interim",
"title": "Solve $$e^{γt}\\left(γ^{2}-γ+121\\right)=0:{\\quad}γ=\\frac{1}{2}+i\\frac{\\sqrt{483}}{2},\\:γ=\\frac{1}{2}-i\\frac{\\sqrt{483}}{2}$$",
"input": "e^{γt}\\left(γ^{2}-γ+121\\right)=0",
"steps": [
{
"type": "step",
"primary": "Since $$e^{γt}\\ne\\:0$$, solving $$e^{γt}\\left(γ^{2}-γ+121\\right)=0$$<br/> is equivalent to solving the quadratic equation $$γ^{2}-γ+121=0$$",
"result": "γ^{2}-γ+121=0"
},
{
"type": "interim",
"title": "Solve with the quadratic formula",
"input": "γ^{2}-γ+121=0",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:1\\cdot\\:121}}{2\\cdot\\:1}",
"steps": [
{
"type": "definition",
"title": "Quadratic Equation Formula:",
"text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$"
},
{
"type": "step",
"primary": "For $${\\quad}a=1,\\:b=-1,\\:c=121$$",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:1\\cdot\\:121}}{2\\cdot\\:1}"
}
],
"meta": {
"interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq",
"gptData": "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"
}
},
{
"type": "interim",
"title": "Simplify $$\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:1\\cdot\\:121}:{\\quad}\\sqrt{483}i$$",
"input": "\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:1\\cdot\\:121}",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{483}i}{2\\cdot\\:1}",
"steps": [
{
"type": "interim",
"title": "$$\\left(-1\\right)^{2}=1$$",
"input": "\\left(-1\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even",
"secondary": [
"$$\\left(-1\\right)^{2}=1^{2}$$"
],
"result": "=1^{2}"
},
{
"type": "step",
"primary": "Apply rule $$1^{a}=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g="
}
},
{
"type": "interim",
"title": "$$4\\cdot\\:1\\cdot\\:121=484$$",
"input": "4\\cdot\\:1\\cdot\\:121",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:121=484$$",
"result": "=484"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7c0qtp6QqZdTxND6Y0QEWHVwkKGJWEPFPk38sdJMsyPI5v4BiYr/j5RJR51+n8IwSMRQ2Rwca5Tm02OJRCslEJ9XABBR49BhKfDbLf0QHU3n"
}
},
{
"type": "step",
"result": "=\\sqrt{1-484}"
},
{
"type": "step",
"primary": "Subtract the numbers: $$1-484=-483$$",
"result": "=\\sqrt{-483}"
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{-1}\\sqrt{a}$$",
"secondary": [
"$$\\sqrt{-483}=\\sqrt{-1}\\sqrt{483}$$"
],
"result": "=\\sqrt{-1}\\sqrt{483}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$",
"result": "=\\sqrt{483}i",
"meta": {
"practiceLink": "/practice/complex-numbers-practice",
"practiceTopic": "Expand FOIL"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"primary": "Separate the solutions",
"result": "{γ}_{1}=\\frac{-\\left(-1\\right)+\\sqrt{483}i}{2\\cdot\\:1},\\:{γ}_{2}=\\frac{-\\left(-1\\right)-\\sqrt{483}i}{2\\cdot\\:1}"
},
{
"type": "interim",
"title": "$$γ=\\frac{-\\left(-1\\right)+\\sqrt{483}i}{2\\cdot\\:1}:{\\quad}\\frac{1}{2}+i\\frac{\\sqrt{483}}{2}$$",
"input": "\\frac{-\\left(-1\\right)+\\sqrt{483}i}{2\\cdot\\:1}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$-\\left(-a\\right)=a$$",
"result": "=\\frac{1+\\sqrt{483}i}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:1=2$$",
"result": "=\\frac{1+\\sqrt{483}i}{2}"
},
{
"type": "interim",
"title": "Rewrite $$\\frac{1+\\sqrt{483}i}{2}$$ in standard complex form: $$\\frac{1}{2}+\\frac{\\sqrt{483}}{2}i$$",
"input": "\\frac{1+\\sqrt{483}i}{2}",
"steps": [
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$",
"secondary": [
"$$\\frac{1+\\sqrt{483}i}{2}=\\frac{1}{2}+\\frac{\\sqrt{483}i}{2}$$"
],
"result": "=\\frac{1}{2}+\\frac{\\sqrt{483}i}{2}"
}
],
"meta": {
"interimType": "Rewrite In Complex Form Title 2Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtJEyuaCK6wsxSwODyDWemAnVQZ9jTQhWaoOGaFYelMKfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZYyz+BRItdDA9ZSSRD/E3DtJEyuaCK6wsxSwODyDWemBMYix7/cTjHR2UPxtgKfavQO9djmXGGOQuWkFSRSz08+WvT9lR6dzrjAd/DNDcBv0="
}
},
{
"type": "step",
"result": "=\\frac{1}{2}+\\frac{\\sqrt{483}}{2}i"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/7pkeZ9PvpPR5geH3JsPMytyDxpoo2WjP0dOy06Ngp9oHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJSQRqTjrUDd23baZ6wFVd3VOypTIYDWwGawep0wQDTvEc93lThIt3Q/3wGfwYj1G2herdkx4fh/64hGjtYuWxjb1FSLpfUk3vbW/M6oOdQCaWNkdcQsCCvOiAG92qWEuDbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "interim",
"title": "$$γ=\\frac{-\\left(-1\\right)-\\sqrt{483}i}{2\\cdot\\:1}:{\\quad}\\frac{1}{2}-i\\frac{\\sqrt{483}}{2}$$",
"input": "\\frac{-\\left(-1\\right)-\\sqrt{483}i}{2\\cdot\\:1}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$-\\left(-a\\right)=a$$",
"result": "=\\frac{1-\\sqrt{483}i}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:1=2$$",
"result": "=\\frac{1-\\sqrt{483}i}{2}"
},
{
"type": "interim",
"title": "Rewrite $$\\frac{1-\\sqrt{483}i}{2}$$ in standard complex form: $$\\frac{1}{2}-\\frac{\\sqrt{483}}{2}i$$",
"input": "\\frac{1-\\sqrt{483}i}{2}",
"steps": [
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$",
"secondary": [
"$$\\frac{1-\\sqrt{483}i}{2}=\\frac{1}{2}-\\frac{\\sqrt{483}i}{2}$$"
],
"result": "=\\frac{1}{2}-\\frac{\\sqrt{483}i}{2}"
}
],
"meta": {
"interimType": "Rewrite In Complex Form Title 2Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYps4fmygri5eHQCBqKksYronVQZ9jTQhWaoOGaFYelMKfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZYyz+BRItdDA9ZSSRD/E3Dps4fmygri5eHQCBqKksYrpMYix7/cTjHR2UPxtgKfavQO9djmXGGOQuWkFSRSz08+WvT9lR6dzrjAd/DNDcBv0="
}
},
{
"type": "step",
"result": "=\\frac{1}{2}-\\frac{\\sqrt{483}}{2}i"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XXvJ86m2ThoGD7P681gRfCtyDxpoo2WjP0dOy06Ngp9oHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJSQRqTjrUDd23baZ6wFVd3X45Pu3hs6dyTvLPVvT6J0Xc93lThIt3Q/3wGfwYj1G2herdkx4fh/64hGjtYuWxjbNrMcvNkQJ7akK/12zrGEiWNkdcQsCCvOiAG92qWEuDbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "step",
"primary": "The solutions to the quadratic equation are:",
"result": "γ=\\frac{1}{2}+i\\frac{\\sqrt{483}}{2},\\:γ=\\frac{1}{2}-i\\frac{\\sqrt{483}}{2}"
}
],
"meta": {
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"result": "γ=\\frac{1}{2}+i\\frac{\\sqrt{483}}{2},\\:γ=\\frac{1}{2}-i\\frac{\\sqrt{483}}{2}"
},
{
"type": "step",
"primary": "For two complex roots $$γ_{1}\\ne\\:γ_{2}$$, where $$γ_{1}=\\alpha+i\\:\\beta,\\:γ_{2}=\\alpha-i\\:\\beta\\:$$<br/>the general solution takes the form:$${\\quad}y=e^{\\alpha\\:t}\\left(c_{1}\\cos\\left(\\beta\\:t\\right)+c_{2}\\sin\\left(\\beta\\:t\\right)\\right)$$",
"result": "e^{\\frac{1}{2}t}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}}{2}t\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}}{2}t\\right)\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)"
}
],
"meta": {
"solvingClass": "ODE",
"interimType": "Generic Find By Solving Title 2Eq"
}
},
{
"type": "interim",
"title": "Find $$y_{p}$$ that satisfies $$y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right):{\\quad}y=\\cos\\left(11t\\right)$$",
"steps": [
{
"type": "step",
"primary": "For the non-homogeneous part $$g\\left(x\\right)=11\\sin\\left(11t\\right)$$, assume a solution of the form: $$y=a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)$$"
},
{
"type": "step",
"result": "\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{^{\\prime\\prime}}-\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{^{\\prime}}+121\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)=11\\sin\\left(11t\\right)"
},
{
"type": "interim",
"title": "Simplify $$\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime\\prime}-\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime}+121\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)=11\\sin\\left(11t\\right):{\\quad}-11a_{0}\\cos\\left(11t\\right)+11a_{1}\\sin\\left(11t\\right)=11\\sin\\left(11t\\right)$$",
"steps": [
{
"type": "interim",
"title": "$$\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime\\prime}=-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)$$",
"input": "\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime\\prime}",
"steps": [
{
"type": "interim",
"title": "$$\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}=a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)$$",
"input": "\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\left(a_{0}\\sin\\left(11t\\right)\\right)^{^{\\prime}}+\\left(a_{1}\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(a_{0}\\sin\\left(11t\\right)\\right)^{\\prime}=a_{0}\\cos\\left(11t\\right)\\cdot\\:11$$",
"input": "\\left(a_{0}\\sin\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{0}\\left(\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\sin\\left(11t\\right)\\right)^{\\prime}",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\sin\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\sin\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"input": "\\left(\\sin\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"result": "=\\cos\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przbRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19InMhwrDgcALtDbcjUqULjO0e5byrQDQVCXUD0vH/fvOdy0Sv9DakosoaSdvz3y7/jiRKGGP17LApxwbbz9SLndow=="
}
},
{
"type": "step",
"result": "=\\cos\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=11t$$",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przdXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcSTIe30CBuWmP1+uyGYZgZDHt0FLgpzbmBtaZEH6JjLtbA+zX4bD3u3gx65o2NJhPKORWtrRqMKFKC4wx/18aolRaVn5uelN6bd2oDGG00ow=="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{0}\\cos\\left(11t\\right)\\cdot\\:11"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\left(a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}=-11a_{1}\\sin\\left(11t\\right)$$",
"input": "\\left(a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{1}\\left(\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\cos\\left(11t\\right)\\right)^{\\prime}",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\cos\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\cos\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"input": "\\left(\\cos\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyLRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19In4/yNG6RtaMeFUKXMVc75z2DpznShoZau34eGycg+7G3mVcoSp6dsHpd9bDcXuJ0pz2gvuHI/drUkZ7NmziNx4SS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=11t$$",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyNXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcOXl4x9OXOWFHQY9rAT64IGgK/z24D94dcZ1AqLbMwBPC30sSftAIFS6Qkpy19IkrNs0uRhmwmWtV92tk+c/8Zu/mDTHcAziAiYeNOzjloJg=="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{1}\\left(-\\sin\\left(11t\\right)\\cdot\\:11\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-11a_{1}\\sin\\left(11t\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)\\right)^{\\prime}=-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)$$",
"input": "\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11\\right)^{^{\\prime}}-\\left(11a_{1}\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11\\right)^{\\prime}=-121a_{0}\\sin\\left(11t\\right)$$",
"input": "\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{0}\\cdot\\:11\\left(\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\cos\\left(11t\\right)\\right)^{\\prime}",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\cos\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\cos\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"input": "\\left(\\cos\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyLRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19In4/yNG6RtaMeFUKXMVc75z2DpznShoZau34eGycg+7G3mVcoSp6dsHpd9bDcXuJ0pz2gvuHI/drUkZ7NmziNx4SS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=11t$$",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyNXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcOXl4x9OXOWFHQY9rAT64IGgK/z24D94dcZ1AqLbMwBPC30sSftAIFS6Qkpy19IkrNs0uRhmwmWtV92tk+c/8Zu/mDTHcAziAiYeNOzjloJg=="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{0}\\cdot\\:11\\left(-\\sin\\left(11t\\right)\\cdot\\:11\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-121a_{0}\\sin\\left(11t\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\left(11a_{1}\\sin\\left(11t\\right)\\right)^{\\prime}=121a_{1}\\cos\\left(11t\\right)$$",
"input": "\\left(11a_{1}\\sin\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11a_{1}\\left(\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\sin\\left(11t\\right)\\right)^{\\prime}",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\sin\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\sin\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"input": "\\left(\\sin\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"result": "=\\cos\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przbRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19InMhwrDgcALtDbcjUqULjO0e5byrQDQVCXUD0vH/fvOdy0Sv9DakosoaSdvz3y7/jiRKGGP17LApxwbbz9SLndow=="
}
},
{
"type": "step",
"result": "=\\cos\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=11t$$",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przdXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcSTIe30CBuWmP1+uyGYZgZDHt0FLgpzbmBtaZEH6JjLtbA+zX4bD3u3gx65o2NJhPKORWtrRqMKFKC4wx/18aolRaVn5uelN6bd2oDGG00ow=="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=11a_{1}\\cos\\left(11t\\right)\\cdot\\:11"
},
{
"type": "step",
"primary": "Simplify",
"result": "=121a_{1}\\cos\\left(11t\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)-\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{^{\\prime}}+121\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)=11\\sin\\left(11t\\right)"
},
{
"type": "interim",
"title": "$$\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime}=a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)$$",
"input": "\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\left(a_{0}\\sin\\left(11t\\right)\\right)^{^{\\prime}}+\\left(a_{1}\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(a_{0}\\sin\\left(11t\\right)\\right)^{\\prime}=a_{0}\\cos\\left(11t\\right)\\cdot\\:11$$",
"input": "\\left(a_{0}\\sin\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{0}\\left(\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\sin\\left(11t\\right)\\right)^{\\prime}",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\sin\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\sin\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"input": "\\left(\\sin\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"result": "=\\cos\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przbRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19InMhwrDgcALtDbcjUqULjO0e5byrQDQVCXUD0vH/fvOdy0Sv9DakosoaSdvz3y7/jiRKGGP17LApxwbbz9SLndow=="
}
},
{
"type": "step",
"result": "=\\cos\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=11t$$",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przdXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcSTIe30CBuWmP1+uyGYZgZDHt0FLgpzbmBtaZEH6JjLtbA+zX4bD3u3gx65o2NJhPKORWtrRqMKFKC4wx/18aolRaVn5uelN6bd2oDGG00ow=="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{0}\\cos\\left(11t\\right)\\cdot\\:11"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\left(a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}=-11a_{1}\\sin\\left(11t\\right)$$",
"input": "\\left(a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{1}\\left(\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Apply the chain rule:$${\\quad}-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\cos\\left(11t\\right)\\right)^{\\prime}",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\cos\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\cos\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"input": "\\left(\\cos\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Apply the common derivative: $$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyLRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19In4/yNG6RtaMeFUKXMVc75z2DpznShoZau34eGycg+7G3mVcoSp6dsHpd9bDcXuJ0pz2gvuHI/drUkZ7NmziNx4SS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Substitute back $$u=11t$$",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyNXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcOXl4x9OXOWFHQY9rAT64IGgK/z24D94dcZ1AqLbMwBPC30sSftAIFS6Qkpy19IkrNs0uRhmwmWtV92tk+c/8Zu/mDTHcAziAiYeNOzjloJg=="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{1}\\left(-\\sin\\left(11t\\right)\\cdot\\:11\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-11a_{1}\\sin\\left(11t\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)-\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)\\right)+121\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)=11\\sin\\left(11t\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "-11a_{0}\\cos\\left(11t\\right)+11a_{1}\\sin\\left(11t\\right)=11\\sin\\left(11t\\right)"
}
],
"meta": {
"interimType": "ODE Derive And Simplify 0Eq"
}
},
{
"type": "step",
"primary": "Find a solution for the coefficient(s) $$a_{0},\\:a_{1}$$"
},
{
"type": "interim",
"title": "Solve $$-11a_{0}\\cos\\left(11t\\right)+11a_{1}\\sin\\left(11t\\right)=11\\sin\\left(11t\\right):{\\quad}a_{0}=0,\\:a_{1}=1$$",
"steps": [
{
"type": "step",
"primary": "Group like terms",
"result": "-11a_{0}\\cos\\left(11t\\right)+11a_{1}\\sin\\left(11t\\right)=11\\sin\\left(11t\\right)"
},
{
"type": "step",
"primary": "Equate the coefficients of similar terms on both sides to create a list of equations",
"result": "\\begin{bmatrix}11=11a_{1}\\\\0=-11a_{0}\\end{bmatrix}"
},
{
"type": "interim",
"title": "Solve system of equations:$${\\quad}a_{0}=0,\\:a_{1}=1$$",
"result": "a_{0}=0,\\:a_{1}=1",
"steps": [
{
"type": "step",
"result": "\\begin{bmatrix}11=11a_{1}\\\\0=-11a_{0}\\end{bmatrix}"
},
{
"type": "interim",
"title": "Isolate $$a_{1}\\:$$for $$11=11a_{1}:{\\quad}a_{1}=1$$",
"input": "11=11a_{1}",
"steps": [
{
"type": "step",
"primary": "Switch sides",
"result": "11a_{1}=11"
},
{
"type": "interim",
"title": "Divide both sides by $$11$$",
"input": "11a_{1}=11",
"result": "a_{1}=1",
"steps": [
{
"type": "step",
"primary": "Divide both sides by $$11$$",
"result": "\\frac{11a_{1}}{11}=\\frac{11}{11}"
},
{
"type": "step",
"primary": "Simplify",
"result": "a_{1}=1"
}
],
"meta": {
"interimType": "Divide Both Sides Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tOqbN2aeKYRdl8kuIY93rgboBkjcWgPxNy7lfBXbA9vya/gmXRIc30tREPIIyab8j4AvVnYTxXcUTFhQ0hFvun7bCV/dLZt6Z1PbdYiokkjFzOlwNoltHDsbCIiHJpcM0hYzF6vxjSZoMSInCY/PPQbO95/2yiZOrh8Vdx/OHPtL0rMteoF5FEvu8HePHkHFm4dyRMR47DyTRgn/YwP6RmYJTz9iY6R2JNTfPX0KV5u2+9vzrtnl6+0mcvSNieKFbdNl//QVuJjXX1DbjBkNuZiWk9GMxTHCKLlF4R6hOQged9TIA/KnqmGBBYRLrKTm0W3AdxChn1fX7F/ZLSj3eUkot7PcZS/Y6l8t49dHfmrWwPs1+Gw97t4MeuaNjSYT8Uf1LNn44VnGjqt9m04yNrGYCvxLr2rzQlp+Ib9xya8="
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Isolate Title 2Eq"
}
},
{
"type": "step",
"result": "\\begin{bmatrix}0=-11a_{0}\\end{bmatrix}"
},
{
"type": "interim",
"title": "Isolate $$a_{0}\\:$$for $$0=-11a_{0}:{\\quad}a_{0}=0$$",
"input": "0=-11a_{0}",
"steps": [
{
"type": "step",
"primary": "Switch sides",
"result": "-11a_{0}=0"
},
{
"type": "interim",
"title": "Divide both sides by $$-11$$",
"input": "-11a_{0}=0",
"result": "a_{0}=0",
"steps": [
{
"type": "step",
"primary": "Divide both sides by $$-11$$",
"result": "\\frac{-11a_{0}}{-11}=\\frac{0}{-11}"
},
{
"type": "step",
"primary": "Simplify",
"result": "a_{0}=0"
}
],
"meta": {
"interimType": "Divide Both Sides Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UDlCeFpQOgo07CJinwRMiIVF+8nruyyk8MUaUUtJFMHya/gmXRIc30tREPIIyab8mTfYzPwngpIq6TMUZSWn2rIYekCLR396I6jWKyllCjUxnb0cY/fOn1kmKV0d1io8cXGfWuywtjebrMbja5E8JJDDsPvhQkQSaykqx9sc+sYtAn6lO7qZ/lL4/WVejrJ5L9Q3ExfflxJSbMXVAtkXbgUJjkSIb5lcKfn2jTZ7xWMVk8t2Twjh+zjp9dlYTco/GgwSbIcd5JjP/d1KuCbW94zDSf2vh/i+3s4cO/oKwZy1rKw+72V41LgxeCOecKXOfIvtRFItCUfO72DblJDvr6ag359nJICSNxR1sdkiJYDwt9LEn7QCBUukJKctfSJK0WcS2OQ+KOcgbSBFF8tALRhsl09fsMPL5dvgr0EoMbk="
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Isolate Title 2Eq"
}
},
{
"type": "step",
"primary": "The solutions to the system of equations are:",
"result": "a_{0}=0,\\:a_{1}=1"
}
],
"meta": {
"solvingClass": "System of Equations",
"interimType": "Partial Fraction Solve System Equation 0Eq"
}
}
],
"meta": {
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"primary": "Plug the parameter solutions into $$y=a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)$$",
"result": "y=0\\cdot\\:\\sin\\left(11t\\right)+1\\cdot\\:\\cos\\left(11t\\right)"
},
{
"type": "step",
"primary": "Simplify",
"result": "y=\\cos\\left(11t\\right)"
},
{
"type": "step",
"primary": "A particular solution $$y_{p}$$ to$${\\quad}y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right){\\quad}$$is:",
"result": "y=\\cos\\left(11t\\right)"
}
],
"meta": {
"interimType": "Generic Find That Satisfies Title 2Eq"
}
},
{
"type": "step",
"primary": "The general solution $$y=y_h+y_p$$ is:",
"result": "y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)+\\cos\\left(11t\\right)"
}
],
"meta": {
"interimType": "ODE Solve Linear 0Eq"
}
},
{
"type": "step",
"result": "y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)+\\cos\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "ODE"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "t",
"plotRequest": "#>#ODE#>#y=e^{\\frac{t}{2}}(c_{1}\\cos(\\frac{\\sqrt{483}t}{2})+c_{2}\\sin(\\frac{\\sqrt{483}t}{2}))+\\cos(11t)"
}
}
},
"meta": {
"showVerify": true
}
}
Solution
Solution
Solution steps
Solve linear ODE: