What is the solution for x^{''}-x=e^{-t}sin(e^{-t})+cos(e^{-t}) ?

The solution for x^{''}-x=e^{-t}sin(e^{-t})+cos(e^{-t}) is x=c_{1}e^t+c_{2}e^{-t}-e^tsin(e^{-t})

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x^{''}-x=e^{-t}sin(e^{-t})+cos(e^{-t})

x^{^{''}} - x = e^{- t} sin (e^{- t}) + cos (e^{- t})

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

Solution steps

x^{''} (t) - x = e^{- t} sin (e^{- t}) + cos (e^{- t})

Solve linear ODE:

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

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

What is the solution for x^{''}-x=e^{-t}sin(e^{-t})+cos(e^{-t}) ?
The solution for x^{''}-x=e^{-t}sin(e^{-t})+cos(e^{-t}) is x=c_{1}e^t+c_{2}e^{-t}-e^tsin(e^{-t})