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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 4 y^{^{'}} - 2 y = 2 t^{2} - 3 t + 6

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 9 y = x e^{x}

y^{^{'''}} - y^{^{'}} = csc (t)

y^{^{'''}} + 4 y^{^{'}} = sin (x) cos (x)

y^{^{'''}} + y^{^{''}} = 3 e^{- x} + x

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