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

The solution for y^{''}-4y=(x-2)e^x is y=c_{1}e^{2x}+c_{2}e^{-2x}-(e^xx)/3+4/9 e^x

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

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

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{e ^{x} x}{3} + \frac{4}{9} e^{x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{e ^{x} x}{3} + \frac{4}{9} e^{x}

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{e ^{x} x}{3} + \frac{4}{9} e^{x}

y^{^{''}} + 4 y^{^{'}} + 3 y = 15

(D^{2} + 1) y = sec^{3} (x)

y^{^{''}} - 4 y^{^{'}} + 3 y = x, y (0) = 6, y^{^{'}} (0) = 10

y^{^{''}} + 6 y^{^{'}} + 25 y = x^{2}

y^{^{''}} + 2 y^{^{'}} - 8 y = - 650 sin (3 t)

What is the solution for y^{''}-4y=(x-2)e^x ?
The solution for y^{''}-4y=(x-2)e^x is y=c_{1}e^{2x}+c_{2}e^{-2x}-(e^xx)/3+4/9 e^x