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

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

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y^{''}-4y=e-2x

y^{^{''}} - 4 y = e - 2 x

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} + \frac{x}{2} - \frac{e}{4}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} + \frac{x}{2} - \frac{e}{4}

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} + \frac{x}{2} - \frac{e}{4}

y^{^{''}} + y^{^{'}} - 2 y = x + sin (2 x), y^{^{'}} (0) = 0, y (0) = 1

y^{^{''}} - 3 y^{^{'}} + 2 y = x^{2} + 8 x + 9

(D^{4} - 8 D^{2} + 16) (y) = x e^{2 x}

\frac{d ^{2}}{d t ^{2}} (θ (t)) + 4 θ (t) = cos^{2} (2 t)

\frac{d ^{3} y}{d x ^{3}} - 12 (\frac{d y}{d x}) + 16 y = 32 x - 8

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