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Popular >

y^{''}+by^'+cy=e^xsin(x)

Solution

y^{^{''}} + b y^{^{'}} + cy = e^{x} sin (x)

Solution

y = c_{1} e^{\frac{x ( - b + b ^{2} - 4 c )}{2}} + c_{2} e^{\frac{x ( - b - b ^{2} - 4 c )}{2}} + \frac{e ^{x} sin ( x ) ( b + c )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b} - \frac{e ^{x} cos ( x ) ( 2 + b )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b}

Solution steps

y^{''} (x) + b y^{'} (x) + cy (x) = e^{x} sin (x)

Solve linear ODE:

y = c_{1} e^{\frac{x ( - b + b ^{2} - 4 c )}{2}} + c_{2} e^{\frac{x ( - b - b ^{2} - 4 c )}{2}} + \frac{e ^{x} sin ( x ) ( b + c )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b} - \frac{e ^{x} cos ( x ) ( 2 + b )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b}

y = c_{1} e^{\frac{x ( - b + b ^{2} - 4 c )}{2}} + c_{2} e^{\frac{x ( - b - b ^{2} - 4 c )}{2}} + \frac{e ^{x} sin ( x ) ( b + c )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b} - \frac{e ^{x} cos ( x ) ( 2 + b )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b}

Graph

Popular Examples

10y^{''}+50y^'+57.6y=cos(x)

10 y^{^{''}} + 50 y^{^{'}} + 57.6 y = cos (x)

y^{''}+y=5sin(4x)

y^{^{''}} + y = 5 sin (4 x)

12y^{''}+5y^'-3y=e^x

12 y^{^{''}} + 5 y^{^{'}} - 3 y = e^{x}

y^{''}-3y^'-2y=9e^{-x}

y^{^{''}} - 3 y^{^{'}} - 2 y = 9 e^{- x}

(D^3+D^2+5D+5)y=5cos(2x)

(D^{3} + D^{2} + 5 D + 5) y = 5 cos (2 x)

Frequently Asked Questions (FAQ)

What is the solution for y^{''}+by^'+cy=e^xsin(x) ?
The solution for y^{''}+by^'+cy=e^xsin(x) is