What is the solution for y^{''}-y^'-2y=(6x^2+8x+7)e^x ?

The solution for y^{''}-y^'-2y=(6x^2+8x+7)e^x is y=c_{1}e^{2x}+c_{2}e^{-x}-3x^2e^x-7e^xx-10e^x

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y^{''}-y^'-2y=(6x^2+8x+7)e^x

y^{^{''}} - y^{^{'}} - 2 y = (6 x^{2} + 8 x + 7) e^{x}

y = c_{1} e^{2 x} + c_{2} e^{- x} - 3 x^{2} e^{x} - 7 e^{x} x - 10 e^{x}

Solution steps

y^{''} (x) - y^{'} (x) - 2 y = (6 x^{2} + 8 x + 7) e^{x}

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- x} - 3 x^{2} e^{x} - 7 e^{x} x - 10 e^{x}

y = c_{1} e^{2 x} + c_{2} e^{- x} - 3 x^{2} e^{x} - 7 e^{x} x - 10 e^{x}

y^{^{''}} - 2 y^{^{'}} - 3 y = (3 x^{2} - 5) e^{- x}

2 y^{^{''}} - 8 y = 16 e^{x}

(d^{3} - 3 d - 2) y = 540 x^{3} e^{- x}

y^{^{''}} + y = 2 sin (2 t), y (0) = 9, y^{^{'}} (0) = 0

y^{^{''}} + 4 y = sec (x)

What is the solution for y^{''}-y^'-2y=(6x^2+8x+7)e^x ?
The solution for y^{''}-y^'-2y=(6x^2+8x+7)e^x is y=c_{1}e^{2x}+c_{2}e^{-x}-3x^2e^x-7e^xx-10e^x