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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{'''}} + 3 y^{^{''}} + 9 y^{^{'}} + 27 y = e^{x} + 3

y^{^{''}} - 2 y^{^{'}} + 10 y = 15 x^{2} - x + 1

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

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

y^{^{''}} - 17 y^{^{'}} + 72 y = 55 + 72 x

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