What is the solution for y^{''}+15y^'+56y=e^{-6x} ?

The solution for y^{''}+15y^'+56y=e^{-6x} is y=c_{1}e^{-7x}+c_{2}e^{-8x}+1/2 e^{-6x}

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y^{''}+15y^'+56y=e^{-6x}

y^{^{''}} + 15 y^{^{'}} + 56 y = e^{- 6 x}

y = c_{1} e^{- 7 x} + c_{2} e^{- 8 x} + \frac{1}{2} e^{- 6 x}

Solution steps

y^{''} (x) + 15 y^{'} (x) + 56 y = e^{- 6 x}

Solve linear ODE:

y = c_{1} e^{- 7 x} + c_{2} e^{- 8 x} + \frac{1}{2} e^{- 6 x}

y = c_{1} e^{- 7 x} + c_{2} e^{- 8 x} + \frac{1}{2} e^{- 6 x}

y^{^{''}} + 4 y = 6 cos (2 x)

y^{^{''}} + 3 y^{^{'}} - 10 y = e^{2 x} + x e^{- 5 x}

y^{^{'''}} - y^{^{''}} - 25 y^{^{'}} + 25 y = 6 - e^{x} + e^{5 x}

y^{^{''''}} + 2 y^{^{''}} + y = 7 + cos (3 t)

y^{^{''}} - 2 y^{^{'}} + y = e^{t}, y (0) = 0, y^{^{'}} (0) = 5

What is the solution for y^{''}+15y^'+56y=e^{-6x} ?
The solution for y^{''}+15y^'+56y=e^{-6x} is y=c_{1}e^{-7x}+c_{2}e^{-8x}+1/2 e^{-6x}