What is the solution for y^{'''}-15y^{''}+27y^'+243y=0 ?

The solution for y^{'''}-15y^{''}+27y^'+243y=0 is y=c_{1}e^{-3t}+c_{2}e^{9t}+c_{3}te^{9t}

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y^{'''}-15y^{''}+27y^'+243y=0

y^{^{'''}} - 15 y^{^{''}} + 27 y^{^{'}} + 243 y = 0

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

Solution steps

y^{'''} (t) - 15 y^{''} (t) + 27 y^{'} (t) + 243 y = 0

Solve linear ODE:

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

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

\frac{d ^{2} y}{d x ^{2}} (1) - \frac{d y}{d x} (5) + 4 y = 0

2 x^{^{''}} + 2 x^{^{'}} + 3 x = 0

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

3 y^{^{''}} + 2 y^{^{'}} + 4 y^{^{''}} + y = 0

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

What is the solution for y^{'''}-15y^{''}+27y^'+243y=0 ?
The solution for y^{'''}-15y^{''}+27y^'+243y=0 is y=c_{1}e^{-3t}+c_{2}e^{9t}+c_{3}te^{9t}