What is the solution for y^{''}-4y=2x-sin(x) ?

The solution for y^{''}-4y=2x-sin(x) is y=c_{1}e^{2x}+c_{2}e^{-2x}-x/2+1/5 sin(x)

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y^{''}-4y=2x-sin(x)

y^{^{''}} - 4 y = 2 x - sin (x)

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

Solution steps

y^{''} (x) - 4 y = 2 x - sin (x)

Solve linear ODE:

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

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

y^{^{''}} + y^{^{'}} - 2 y = 5

\frac{d ^{2} y}{d x ^{2}} + 9 y = x e^{2 x} + sin (3 x)

y^{^{''}} - y^{^{'}} - 12 y = 5 - 2 x

2 \frac{d ^{2} y}{d t ^{2}} + 3 \frac{d y}{d t} + y = H (t)

y^{^{''}} - 3 y^{^{'}} - 46 = 2 sin (t)

What is the solution for y^{''}-4y=2x-sin(x) ?
The solution for y^{''}-4y=2x-sin(x) is y=c_{1}e^{2x}+c_{2}e^{-2x}-x/2+1/5 sin(x)