What is the solution for (D^2+D-2)y=e^x ?

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

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(D^2+D-2)y=e^x

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 8 y^{^{'}} + 15 y = e^{3 x} sin (3 x)

x^{^{''}} + a^{2} x = a^{2} b + c

y^{^{''}} - 10 y^{^{'}} + 9 y = 5

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 78 x e^{x}

y^{^{''}} + y^{^{'}} - 12 y = 2 cos (t)

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