What is the solution for (d^2y)/(dx^2)-y=e^x ?

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

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(d^2y)/(dx^2)-y=e^x

\frac{d ^{2} y}{d x ^{2}} - y = e^{x}

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

Solution steps

\frac{d ^{2} y}{d x ^{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}

y^{^{''}} + 25 y^{^{'}} = - 24 t e^{- t}

y^{^{''}} + 4 y^{^{'}} + y = (\frac{e ^{- \frac{x}{2}}}{x ^{2}})

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

2 x^{^{''}} + 3 x^{^{'}} + 4 x = - 5 cos (t) + sin (t)

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

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