What is the solution for x^{''}-18x^'+81x=3te^{9t} ?

The solution for x^{''}-18x^'+81x=3te^{9t} is x=c_{1}e^{9t}+c_{2}te^{9t}+(e^{9t}t^3)/2

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x^{''}-18x^'+81x=3te^{9t}

x^{^{''}} - 18 x^{^{'}} + 81 x = 3 t e^{9 t}

x = c_{1} e^{9 t} + c_{2} t e^{9 t} + \frac{e ^{9 t} t ^{3}}{2}

Solution steps

x^{''} (t) - 18 x^{'} (t) + 81 x = 3 t e^{9 t}

Solve linear ODE:

x = c_{1} e^{9 t} + c_{2} t e^{9 t} + \frac{e ^{9 t} t ^{3}}{2}

x = c_{1} e^{9 t} + c_{2} t e^{9 t} + \frac{e ^{9 t} t ^{3}}{2}

y^{^{''}} + 2 y^{^{'}} + y = 2 x^{2} + 4 x + 2

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

y^{^{''}} - 8 y^{^{'}} + 16 y = t^{- 5} e^{4 t}

x^{^{''}} + 7 x^{^{'}} + 12 x = e^{2 t} + cos (4 t)

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

What is the solution for x^{''}-18x^'+81x=3te^{9t} ?
The solution for x^{''}-18x^'+81x=3te^{9t} is x=c_{1}e^{9t}+c_{2}te^{9t}+(e^{9t}t^3)/2