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

The solution for y^{''}-2y^'+y=x^2-x-3 is y=c_{1}e^x+c_{2}xe^x+x^2+3x+1

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y^{''}-2y^'+y=x^2-x-3

y^{^{''}} - 2 y^{^{'}} + y = x^{2} - x - 3

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

Solution steps

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

Solve linear ODE:

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

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

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