What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=3x^2-5 ?

The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=3x^2-5 is y=c_{1}e^x+c_{2}xe^x+3x^2+12x+13

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(d^2y)/(dx^2)-2(dy)/(dx)+y=3x^2-5

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = 3 x^{2} - 5

y = c_{1} e^{x} + c_{2} x e^{x} + 3 x^{2} + 12 x + 13

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = 3 x^{2} - 5

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + 3 x^{2} + 12 x + 13

y = c_{1} e^{x} + c_{2} x e^{x} + 3 x^{2} + 12 x + 13

4 y^{^{''}} + 36 y = csc (3 x)

y^{^{''}} - y = cosh (x)

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

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

4 y^{^{''}} - 4 y^{^{'}} + y = e^{\frac{x}{2}} 1 - x^{2}

What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=3x^2-5 ?
The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=3x^2-5 is y=c_{1}e^x+c_{2}xe^x+3x^2+12x+13