What is the solution for x^{''}-4x^'+4x=84t^2e^{2t} ?

The solution for x^{''}-4x^'+4x=84t^2e^{2t} is x=c_{1}e^{2t}+c_{2}te^{2t}+7e^{2t}t^4

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x^{''}-4x^'+4x=84t^2e^{2t}

x^{^{''}} - 4 x^{^{'}} + 4 x = 84 t^{2} e^{2 t}

x = c_{1} e^{2 t} + c_{2} t e^{2 t} + 7 e^{2 t} t^{4}

Solution steps

x^{''} (t) - 4 x^{'} (t) + 4 x = 84 t^{2} e^{2 t}

Solve linear ODE:

x = c_{1} e^{2 t} + c_{2} t e^{2 t} + 7 e^{2 t} t^{4}

x = c_{1} e^{2 t} + c_{2} t e^{2 t} + 7 e^{2 t} t^{4}

y^{^{''}} + 25 y = 25, y (0) = 1, y^{^{'}} (0) = 5

y^{^{''}} - 2 y^{^{'}} - 3 y = e^{4 x}

y^{^{''}} + 10 y^{^{'}} + 25 y = - 8 - 5 x + 25 x^{2}

10 x^{^{''}} + 200 x = 600 sin (125 t)

5 y^{^{''}} + 6 y^{^{'}} + y = x^{3} + 18 x^{2} + 32 x + 13

What is the solution for x^{''}-4x^'+4x=84t^2e^{2t} ?
The solution for x^{''}-4x^'+4x=84t^2e^{2t} is x=c_{1}e^{2t}+c_{2}te^{2t}+7e^{2t}t^4