What is the solution for y^{''}-a*y^'=b*x^2 ?

The solution for y^{''}-a*y^'=b*x^2 is y=c_{1}e^{ax}+c_{2}-(bx^3)/(3a)-(bx^2)/(a^2)-(2bx)/(a^3)

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y^{''}-ay^'=bx^2

y^{^{''}} - a \cdot y^{^{'}} = b \cdot x^{2}

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

Solution steps

y^{''} (x) - a \cdot y^{'} (x) = b \cdot x^{2}

Solve linear ODE:

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

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

y^{^{''}} + 6 y^{^{'}} + 2 y = e^{- 4 t}

y^{^{''}} + 4 y + 5 y = e^{- t}

y^{^{''}} - 6 y^{^{'}} + 9 y = t^{2} \cdot e^{3 t}

y^{^{''}} - 36 y = 9 e^{6 x}

\frac{d ^{2} y}{d x ^{2}} + 4 y = 8

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