What is the solution for y^{'''}+3y^{''}-54y^'=0 ?

The solution for y^{'''}+3y^{''}-54y^'=0 is y=c_{1}+c_{2}e^{6t}+c_{3}e^{-9t}

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y^{'''}+3y^{''}-54y^'=0

y^{^{'''}} + 3 y^{^{''}} - 54 y^{^{'}} = 0

y = c_{1} + c_{2} e^{6 t} + c_{3} e^{- 9 t}

Solution steps

y^{'''} (t) + 3 y^{''} (t) - 54 y^{'} (t) = 0

Solve linear ODE:

y = c_{1} + c_{2} e^{6 t} + c_{3} e^{- 9 t}

y = c_{1} + c_{2} e^{6 t} + c_{3} e^{- 9 t}

\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} + y = 0

5 y^{^{''}} - 6 y^{^{'}} + y = 0, y (0) = 5, y^{^{'}} (0) = \frac{1}{5}

x^{^{''}} + 4 x^{^{'}} + 8 x = 0

9 y^{^{''}} + 25 y = 0

y^{^{''}} - b y = 0

What is the solution for y^{'''}+3y^{''}-54y^'=0 ?
The solution for y^{'''}+3y^{''}-54y^'=0 is y=c_{1}+c_{2}e^{6t}+c_{3}e^{-9t}