What is the solution for y^{''}-y=4e^t-2e^{-t},y(0)=1,y^'(0)=4 ?

The solution for y^{''}-y=4e^t-2e^{-t},y(0)=1,y^'(0)=4 is y=2e^tt+e^{-t}t+e^t

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y^{''}-y=4e^t-2e^{-t},y(0)=1,y^'(0)=4

y^{^{''}} - y = 4 e^{t} - 2 e^{- t}, y (0) = 1, y^{^{'}} (0) = 4

y = 2 e^{t} t + e^{- t} t + e^{t}

Solution steps

y^{''} (t) - y = 4 e^{t} - 2 e^{- t}

Solve linear ODE:

y = 2 e^{t} t + e^{- t} t + e^{t}

y = 2 e^{t} t + e^{- t} t + e^{t}

(D^{4} - 1) y = x

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

y^{^{''}} - 4 y = e - 2 x

y^{^{''}} + y^{^{'}} - 2 y = x + sin (2 x), y^{^{'}} (0) = 0, y (0) = 1

y^{^{''}} - 3 y^{^{'}} + 2 y = x^{2} + 8 x + 9

What is the solution for y^{''}-y=4e^t-2e^{-t},y(0)=1,y^'(0)=4 ?
The solution for y^{''}-y=4e^t-2e^{-t},y(0)=1,y^'(0)=4 is y=2e^tt+e^{-t}t+e^t