What is the solution for -2y^{''}+7y^'-y-8=-6x^2-2x^3e^x ?

The solution for -2y^{''}+7y^'-y-8=-6x^2-2x^3e^x is

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-2y^{''}+7y^'-y-8=-6x^2-2x^3e^x

- 2 y^{^{''}} + 7 y^{^{'}} - y - 8 = - 6 x^{2} - 2 x^{3} e^{x}

y = c_{1} e^{\frac{( 7 - 41 ) x}{4}} + c_{2} e^{\frac{( 7 + 41 ) x}{4}} - \frac{e ^{x} x ^{3}}{2} + 6 x^{2} + \frac{9 e ^{x} x ^{2}}{8} + 84 x - \frac{51 e ^{x} x}{16} + \frac{225}{64} e^{x} + 556

Solution steps

- 2 y^{''} (x) + 7 y^{'} (x) - y - 8 = - 6 x^{2} - 2 x^{3} e^{x}

Solve linear ODE:

y = c_{1} e^{\frac{( 7 - 41 ) x}{4}} + c_{2} e^{\frac{( 7 + 41 ) x}{4}} - \frac{e ^{x} x ^{3}}{2} + 6 x^{2} + \frac{9 e ^{x} x ^{2}}{8} + 84 x - \frac{51 e ^{x} x}{16} + \frac{225}{64} e^{x} + 556

y = c_{1} e^{\frac{( 7 - 41 ) x}{4}} + c_{2} e^{\frac{( 7 + 41 ) x}{4}} - \frac{e ^{x} x ^{3}}{2} + 6 x^{2} + \frac{9 e ^{x} x ^{2}}{8} + 84 x - \frac{51 e ^{x} x}{16} + \frac{225}{64} e^{x} + 556

What is the solution for -2y^{''}+7y^'-y-8=-6x^2-2x^3e^x ?
The solution for -2y^{''}+7y^'-y-8=-6x^2-2x^3e^x is