What is the solution for x^{'''}+x^{''}-72x^'=0 ?

The solution for x^{'''}+x^{''}-72x^'=0 is x=c_{1}+c_{2}e^{8t}+c_{3}e^{-9t}

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x^{'''}+x^{''}-72x^'=0

x^{^{'''}} + x^{^{''}} - 72 x^{^{'}} = 0

x = c_{1} + c_{2} e^{8 t} + c_{3} e^{- 9 t}

Solution steps

x^{'''} (t) + x^{''} (t) - 72 x^{'} (t) = 0

Solve linear ODE:

x = c_{1} + c_{2} e^{8 t} + c_{3} e^{- 9 t}

x = c_{1} + c_{2} e^{8 t} + c_{3} e^{- 9 t}

y^{^{''''}} = 256 y

\frac{1}{2} y^{^{''}} - y^{^{'}} = 0

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

2 y^{^{''}} - 3 y^{^{'}} + 8 y = 0

y^{^{''}} + 14 y^{^{'}} + 49 y = 0, y (0) = - 1, y^{^{'}} (0) = 1

What is the solution for x^{'''}+x^{''}-72x^'=0 ?
The solution for x^{'''}+x^{''}-72x^'=0 is x=c_{1}+c_{2}e^{8t}+c_{3}e^{-9t}