What is the solution for 5y^{''}+50y^'+45y=0,y(0)=-8,y^'(0)=0 ?

The solution for 5y^{''}+50y^'+45y=0,y(0)=-8,y^'(0)=0 is y=-9e^{-t}+e^{-9t}

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5y^{''}+50y^'+45y=0,y(0)=-8,y^'(0)=0

5 y^{^{''}} + 50 y^{^{'}} + 45 y = 0, y (0) = - 8, y^{^{'}} (0) = 0

y = - 9 e^{- t} + e^{- 9 t}

Solution steps

5 y^{''} (t) + 50 y^{'} (t) + 45 y = 0

Solve linear ODE:

y = - 9 e^{- t} + e^{- 9 t}

y = - 9 e^{- t} + e^{- 9 t}

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

y^{^{''}} + 4 y = - 4 y^{^{'}}

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

y^{^{''}} + \frac{16}{9} y = 0

u, \frac{d ^{2} u}{d x ^{2}} = u

What is the solution for 5y^{''}+50y^'+45y=0,y(0)=-8,y^'(0)=0 ?
The solution for 5y^{''}+50y^'+45y=0,y(0)=-8,y^'(0)=0 is y=-9e^{-t}+e^{-9t}