What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=x^2-1 ?

The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=x^2-1 is y=c_{1}e^x+c_{2}xe^x+x^2+4x+5

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(d^2y)/(dx^2)-2(dy)/(dx)+y=x^2-1

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = x^{2} - 1

y = c_{1} e^{x} + c_{2} x e^{x} + x^{2} + 4 x + 5

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = x^{2} - 1

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + x^{2} + 4 x + 5

y = c_{1} e^{x} + c_{2} x e^{x} + x^{2} + 4 x + 5

y^{^{''}} - 4 y^{^{'}} + 3 y = 6

2 y^{^{''}} - 5 y^{^{'}} - 3 y = 3 - x

y^{^{''}} - 3 y^{^{'}} + 2 y = 4 t + e^{3 t}

2 y^{^{''}} - 8 y^{^{'}} - 4 y = - 5 x^{2} - 12 x e^{- x}

y^{^{''}} - 5 y^{^{'}} - 24 y = (t + 1) e^{t}

What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=x^2-1 ?
The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=x^2-1 is y=c_{1}e^x+c_{2}xe^x+x^2+4x+5