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

The solution for y^{''}-2y^'+y=x^3+3x is y=c_{1}e^x+c_{2}xe^x+x^3+6x^2+21x+30

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

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

y = c_{1} e^{x} + c_{2} x e^{x} + x^{3} + 6 x^{2} + 21 x + 30

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + x^{3} + 6 x^{2} + 21 x + 30

y = c_{1} e^{x} + c_{2} x e^{x} + x^{3} + 6 x^{2} + 21 x + 30

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

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

y^{^{''}} - 4 y = \frac{x + 5 e ^{2 x}}{e ^{2 x}}

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

y^{^{''}} + 6 y^{^{'}} + 34 y = 68

What is the solution for y^{''}-2y^'+y=x^3+3x ?
The solution for y^{''}-2y^'+y=x^3+3x is y=c_{1}e^x+c_{2}xe^x+x^3+6x^2+21x+30