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

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

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

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

y = c_{1} e^{x} + c_{2} e^{- x} - sin (e^{- x}) - e^{x} cos (e^{- x})

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} - sin (e^{- x}) - e^{x} cos (e^{- x})

y = c_{1} e^{x} + c_{2} e^{- x} - sin (e^{- x}) - e^{x} cos (e^{- x})

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 91 x e^{x}

x^{^{''}} + x = sin^{2} (t)

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

x^{^{''}} + 10 x^{^{'}} + 29 x = t e^{- t}

y^{^{'''}} - y^{^{'}} = 9 t

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