What is the solution for y^{''}-2y^'+y=(x+1)e^x ?

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

2 y^{^{''}} - 2 y^{^{'}} - 4 y = 2 e^{2 x}

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

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

y^{^{''}} - 2 y^{^{'}} + y = \frac{e ^{t}}{( 1 + t ^{2} )}

D (x) = - 6 x + 4200

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