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

The solution for y^{''}-y^'=x^2+2x+3 is y=c_{1}e^x+c_{2}-(x^3)/3-2x^2-7x

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

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

y = c_{1} e^{x} + c_{2} - \frac{x ^{3}}{3} - 2 x^{2} - 7 x

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} - \frac{x ^{3}}{3} - 2 x^{2} - 7 x

y = c_{1} e^{x} + c_{2} - \frac{x ^{3}}{3} - 2 x^{2} - 7 x

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} - 32 y = 6 + 3 x

y^{^{''}} + a y^{^{'}} - 2 a^{2} y = 1

y^{^{''}} + 2 y^{^{'}} + y = e^{- x}, y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} + 6 y^{^{'}} + 9 y = 2 e^{(- 3 x)} \frac{1}{x ^{2} + 1}

y^{^{''}} + 4 y^{^{'}} - 5 y = (- 4 x^{2} - 8 x - 3) e^{3 x}

What is the solution for y^{''}-y^'=x^2+2x+3 ?
The solution for y^{''}-y^'=x^2+2x+3 is y=c_{1}e^x+c_{2}-(x^3)/3-2x^2-7x