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

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

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

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

x = c_{1} e^{3 t} + c_{2} e^{- 2 t} + \frac{1}{3} e^{4 t}

Solution steps

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

Solve linear ODE:

x = c_{1} e^{3 t} + c_{2} e^{- 2 t} + \frac{1}{3} e^{4 t}

x = c_{1} e^{3 t} + c_{2} e^{- 2 t} + \frac{1}{3} e^{4 t}

y^{^{''}} + a y = b

y^{^{''}} + 3 y^{^{'}} + 2 y = e^{2 x}

y^{^{''}} + 25 y = 3 x, y (0) = 1, y^{^{'}} (0) = 5

y^{^{'''}} + 7 y^{^{''}} + 15 y^{^{'}} + 9 = 1

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

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