What is the solution for y^{''}-3y=e^{-2x} ?

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

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

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

y = c_{1} e^{3 x} + c_{2} e^{- 3 x} + e^{- 2 x}

Solution steps

y^{''} (x) - 3 y = e^{- 2 x}

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} e^{- 3 x} + e^{- 2 x}

y = c_{1} e^{3 x} + c_{2} e^{- 3 x} + e^{- 2 x}

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

y^{^{''}} + (9 - 2) y^{^{'}} - 2 (9) y = 8 e^{2 x}

y^{^{''}} + 4 y = 5 sin (3 x)

y^{^{''}} + y - \frac{π}{2} = 0

y^{^{''''}} - 2 y^{^{''}} + y = 100 cos (3 x)

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