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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

\int \frac{x ^{2} + 3}{x ^{3} + 5 x} d x

\frac{d}{d t} (A e^{- t})

\int e^{π} d x

\frac{d}{d x} (- 16 e^{- x^{4}} x^{3})

\int_{0}^{1} \int_{0}^{y^{2}} x e^{\frac{x}{y ^{2}}} d x d y

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