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

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

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y^{''}-4y=e^{-αx}

y^{^{''}} - 4 \cdot y = e^{- α \cdot x}

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

Solution steps

y^{''} (x) - 4 \cdot y = e^{- α \cdot x}

Solve linear ODE:

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

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

y^{^{'''}} + 3 y^{^{''}} + 2 y^{^{'}} = (x^{2} + 4 x + 8) e^{2} x

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

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

y^{^{''}} - 3 y^{^{'}} + 2 y = - e^{t} + 2 e^{2 t}

y^{^{''}} + y = 3 t^{2}, y (0) = 0, y^{^{'}} (0) = 0

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