What is the solution for D(D+3)y=x(5+e^x) ?

The solution for D(D+3)y=x(5+e^x) is y=c_{1}+c_{2}e^{-3x}+(5x^2)/6-(5x)/9+(e^xx)/4-5/16 e^x

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D(D+3)y=x(5+e^x)

D (D + 3) y = x (5 + e^{x})

y = c_{1} + c_{2} e^{- 3 x} + \frac{5 x ^{2}}{6} - \frac{5 x}{9} + \frac{e ^{x} x}{4} - \frac{5}{16} e^{x}

Solution steps

D (D + 3) y = x (5 + e^{x})

Solve linear ODE:

y = c_{1} + c_{2} e^{- 3 x} + \frac{5 x ^{2}}{6} - \frac{5 x}{9} + \frac{e ^{x} x}{4} - \frac{5}{16} e^{x}

y = c_{1} + c_{2} e^{- 3 x} + \frac{5 x ^{2}}{6} - \frac{5 x}{9} + \frac{e ^{x} x}{4} - \frac{5}{16} e^{x}

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

y^{^{''}} - y = e^{t} cos (t), y (0) = 0, y^{^{'}} (0) = 0

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

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

y^{^{''}} + y^{^{'}} = 10 x^{4} + 2

What is the solution for D(D+3)y=x(5+e^x) ?
The solution for D(D+3)y=x(5+e^x) is y=c_{1}+c_{2}e^{-3x}+(5x^2)/6-(5x)/9+(e^xx)/4-5/16 e^x