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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

5 y^{^{''}} + 3 y^{^{'}} + 6 y = 0, y (0) = - 4, y^{^{'}} (0) = 0

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

y^{^{''}} - 9 y = 0, y (0) = 36, y^{^{'}} (0) = - 36

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} + 6 y = 0

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

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