What is the solution for y^{''}-y^'-2y=5e^{4x}+6x ?

The solution for y^{''}-y^'-2y=5e^{4x}+6x is y=c_{1}e^{2x}+c_{2}e^{-x}+1/2 e^{4x}-3x+3/2

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y^{''}-y^'-2y=5e^{4x}+6x

y^{^{''}} - y^{^{'}} - 2 y = 5 e^{4 x} + 6 x

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

Solution steps

y^{''} (x) - y^{'} (x) - 2 y = 5 e^{4 x} + 6 x

Solve linear ODE:

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

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

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = x + e^{- 3 x}

y^{^{''}} + 4 y - 5 = 0

y^{^{''}} - 8 y^{^{'}} - 20 y = 270 e^{x}

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

y^{^{''}} + 10 y^{^{'}} + 25 y = x, y (0) = 0, y^{^{'}} (0) = 0

What is the solution for y^{''}-y^'-2y=5e^{4x}+6x ?
The solution for y^{''}-y^'-2y=5e^{4x}+6x is y=c_{1}e^{2x}+c_{2}e^{-x}+1/2 e^{4x}-3x+3/2