What is the solution for y^{''}-y=e^xsin(x) ?

The solution for y^{''}-y=e^xsin(x) is y=c_{1}e^x+c_{2}e^{-x}-1/5 e^xsin(x)-2/5 e^xcos(x)

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y^{''}-y=e^xsin(x)

y^{^{''}} - y = e^{x} sin (x)

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{1}{5} e^{x} sin (x) - \frac{2}{5} e^{x} cos (x)

Solution steps

y^{''} (x) - y = e^{x} sin (x)

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{1}{5} e^{x} sin (x) - \frac{2}{5} e^{x} cos (x)

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{1}{5} e^{x} sin (x) - \frac{2}{5} e^{x} cos (x)

4 y^{^{''}} - 4 y^{^{'}} + y = 16 e^{(\frac{x}{2})}

\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} - y = 20 x e^{x}

x^{^{''}} + 3 x^{^{'}} - 10 x = 6 e^{4 t}

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} + 7 y = e^{5 x}

y^{^{''}} - 4 y^{^{'}} + 3 y = 2 x

What is the solution for y^{''}-y=e^xsin(x) ?
The solution for y^{''}-y=e^xsin(x) is y=c_{1}e^x+c_{2}e^{-x}-1/5 e^xsin(x)-2/5 e^xcos(x)