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})

y^{^{''}} - 2 y^{^{'}} - 3 y = 2 e x - 10

y^{^{'''}} + 3 y^{^{''}} + y^{^{'}} - 5 y = - 5 x

y^{^{''}} + 4 y = 4 sin (2 t) + 8 cos (2 t)

y^{^{''}} + 4 y = 2 cos (x)

y^{^{''}} + 7 y = 5 t^{2} - 5, y (0) = 0, y^{^{'}} (0) = - 2

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})