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

The solution for y^{''}+2y^'=x^2+e^{-2x} is y=c_{1}+c_{2}e^{-2x}+(x^3)/6+(-x^2+x)/4-(e^{-2x}x)/2

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

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

y = c_{1} + c_{2} e^{- 2 x} + \frac{x ^{3}}{6} + \frac{- x ^{2} + x}{4} - \frac{e ^{- 2 x} x}{2}

Solution steps

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

Solve linear ODE:

y = c_{1} + c_{2} e^{- 2 x} + \frac{x ^{3}}{6} + \frac{- x ^{2} + x}{4} - \frac{e ^{- 2 x} x}{2}

y = c_{1} + c_{2} e^{- 2 x} + \frac{x ^{3}}{6} + \frac{- x ^{2} + x}{4} - \frac{e ^{- 2 x} x}{2}

f (x) = x, at x = 64.2

x \to 3 lim (6 x^{2} + 2)

\int \frac{( x + 1 )}{x ^{2} + 2 x - 4} d x

\int_{1}^{100} \frac{25}{x ^{3}} d x

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What is the solution for y^{''}+2y^'=x^2+e^{-2x} ?
The solution for y^{''}+2y^'=x^2+e^{-2x} is y=c_{1}+c_{2}e^{-2x}+(x^3)/6+(-x^2+x)/4-(e^{-2x}x)/2