What is the solution for y^{''}+6y^'+9y=50e^{2x} ?

The solution for y^{''}+6y^'+9y=50e^{2x} is y=c_{1}e^{-3x}+c_{2}xe^{-3x}+2e^{2x}

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y^{''}+6y^'+9y=50e^{2x}

y^{^{''}} + 6 y^{^{'}} + 9 y = 50 e^{2 x}

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

Solution steps

y^{''} (x) + 6 y^{'} (x) + 9 y = 50 e^{2 x}

Solve linear ODE:

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

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

\frac{d ^{2} y}{d x ^{2}} + 4 \frac{d y}{d x} + 5 y = x \cdot e^{2 x}

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

y^{^{''}} - y^{^{'}} = - 6

y^{^{''}} - 4 y^{^{'}} + 4 y = (x + 1) e^{x}

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

What is the solution for y^{''}+6y^'+9y=50e^{2x} ?
The solution for y^{''}+6y^'+9y=50e^{2x} is y=c_{1}e^{-3x}+c_{2}xe^{-3x}+2e^{2x}