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Popular >

y^{''}+y^'+y2=sin(x^2)

Solution

y^{^{''}} + y^{^{'}} + y 2 = sin (x^{2})

Solution

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{7 x}{2}) + c_{2} sin (\frac{7 x}{2})) + e^{- \frac{x}{2}} cos (\frac{7 x}{2}) \cdot \int - \frac{2 7 e ^{\frac{x}{2}} sin ( \frac{7 x}{2} ) sin ( x ^{2} )}{7} d x + e^{- \frac{x}{2}} sin (\frac{7 x}{2}) \cdot \int \frac{2 7 e ^{\frac{x}{2}} cos ( \frac{7 x}{2} ) sin ( x ^{2} )}{7} d x

Solution steps

y^{''} (x) + y^{'} (x) + y 2 = sin (x^{2})

Solve linear ODE:

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{7 x}{2}) + c_{2} sin (\frac{7 x}{2})) + e^{- \frac{x}{2}} cos (\frac{7 x}{2}) \cdot \int - \frac{2 7 e ^{\frac{x}{2}} sin ( \frac{7 x}{2} ) sin ( x ^{2} )}{7} d x + e^{- \frac{x}{2}} sin (\frac{7 x}{2}) \cdot \int \frac{2 7 e ^{\frac{x}{2}} cos ( \frac{7 x}{2} ) sin ( x ^{2} )}{7} d x

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{7 x}{2}) + c_{2} sin (\frac{7 x}{2})) + e^{- \frac{x}{2}} cos (\frac{7 x}{2}) \cdot \int - \frac{2 7 e ^{\frac{x}{2}} sin ( \frac{7 x}{2} ) sin ( x ^{2} )}{7} d x + e^{- \frac{x}{2}} sin (\frac{7 x}{2}) \cdot \int \frac{2 7 e ^{\frac{x}{2}} cos ( \frac{7 x}{2} ) sin ( x ^{2} )}{7} d x

Popular Examples

10x^{''}+45x^'+2500x=180cos(7pi)

10 x^{^{''}} + 45 x^{^{'}} + 2500 x = 180 cos (7 π)

y^{''}+7y^'+12y=5x

y^{^{''}} + 7 y^{^{'}} + 12 y = 5 x

(d^2y)/(dx^2)-16(dy)/(dx)-64y=7e^{8x}

\frac{d ^{2} y}{d x ^{2}} - 16 \frac{d y}{d x} - 64 y = 7 e^{8 x}

y^{''}-11y^'+24y=2,y(0)=0,y^'(0)=1

y^{^{''}} - 11 y^{^{'}} + 24 y = 2, y (0) = 0, y^{^{'}} (0) = 1

y^{''}+2y^'+10y=10x+2

y^{^{''}} + 2 y^{^{'}} + 10 y = 10 x + 2

Frequently Asked Questions (FAQ)

What is the solution for y^{''}+y^'+y2=sin(x^2) ?
The solution for y^{''}+y^'+y2=sin(x^2) is