What is the solution for x^{''}+4x^'+4x=4t^2+6e^t ?

The solution for x^{''}+4x^'+4x=4t^2+6e^t is x=c_{1}e^{-2t}+c_{2}te^{-2t}+t^2-2t+3/2+2/3 e^t

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x^{''}+4x^'+4x=4t^2+6e^t

x^{^{''}} + 4 x^{^{'}} + 4 x = 4 t^{2} + 6 e^{t}

x = c_{1} e^{- 2 t} + c_{2} t e^{- 2 t} + t^{2} - 2 t + \frac{3}{2} + \frac{2}{3} e^{t}

Solution steps

x^{''} (t) + 4 x^{'} (t) + 4 x = 4 t^{2} + 6 e^{t}

Solve linear ODE:

x = c_{1} e^{- 2 t} + c_{2} t e^{- 2 t} + t^{2} - 2 t + \frac{3}{2} + \frac{2}{3} e^{t}

x = c_{1} e^{- 2 t} + c_{2} t e^{- 2 t} + t^{2} - 2 t + \frac{3}{2} + \frac{2}{3} e^{t}

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

\frac{d ^{2} y}{d x ^{2}} - 16 \frac{d y}{d x} + 16 y = 16 (x - 1)

y^{^{''}} + 9 y = 9 sec^{2} (3 x)

y^{^{''}} + 2 y^{^{'}} + 5 y = 8 e^{- t} cos (2 t)

y^{^{''}} + 4 y = 5 t^{2} e^{t}

What is the solution for x^{''}+4x^'+4x=4t^2+6e^t ?
The solution for x^{''}+4x^'+4x=4t^2+6e^t is x=c_{1}e^{-2t}+c_{2}te^{-2t}+t^2-2t+3/2+2/3 e^t