What is the solution for y^{''}-2y^'+y=8e^x-1/6 x^3+3x ?

The solution for y^{''}-2y^'+y=8e^x-1/6 x^3+3x is y=c_{1}e^x+c_{2}xe^x+4x^2e^x-(x^3)/6-x^2+2

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y^{''}-2y^'+y=8e^x-1/6 x^3+3x

y^{^{''}} - 2 y^{^{'}} + y = 8 e^{x} - \frac{1}{6} x^{3} + 3 x

y = c_{1} e^{x} + c_{2} x e^{x} + 4 x^{2} e^{x} - \frac{x ^{3}}{6} - x^{2} + 2

Solution steps

y^{''} (x) - 2 y^{'} (x) + y = 8 e^{x} - \frac{1}{6} x^{3} + 3 x

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + 4 x^{2} e^{x} - \frac{x ^{3}}{6} - x^{2} + 2

y = c_{1} e^{x} + c_{2} x e^{x} + 4 x^{2} e^{x} - \frac{x ^{3}}{6} - x^{2} + 2

y^{^{''}} + 9 y^{^{'}} - 22 y = e^{- 3 x}

y^{^{''}} + y^{^{'}} = 5

3 y^{^{''}} + 4 y + 2 = 0

y^{^{''}} + 5 y^{^{'}} + 6 y = 2 e^{- t}

\frac{d ^{2} y}{d x ^{2}} + y = tan (x)

What is the solution for y^{''}-2y^'+y=8e^x-1/6 x^3+3x ?
The solution for y^{''}-2y^'+y=8e^x-1/6 x^3+3x is y=c_{1}e^x+c_{2}xe^x+4x^2e^x-(x^3)/6-x^2+2