What is the solution for y^{''}+5y^'+4y=0,y(0)=7,y^'(0)=-13 ?

The solution for y^{''}+5y^'+4y=0,y(0)=7,y^'(0)=-13 is y=5e^{-t}+2e^{-4t}

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y^{''}+5y^'+4y=0,y(0)=7,y^'(0)=-13

y^{^{''}} + 5 y^{^{'}} + 4 y = 0, y (0) = 7, y^{^{'}} (0) = - 13

y = 5 e^{- t} + 2 e^{- 4 t}

Solution steps

y^{''} (t) + 5 y^{'} (t) + 4 y = 0

Solve linear ODE:

y = 5 e^{- t} + 2 e^{- 4 t}

y = 5 e^{- t} + 2 e^{- 4 t}

y^{^{'''}} - 3 y^{^{''}} + 5 y^{^{'}} + 9 y = 0

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

\frac{x ^{^{''}} - 3 x ^{^{'}}}{2} = 2 x - \frac{( x ^{^{'}} - 3 x )}{2}

y^{^{''}} + y^{^{'}} - 12 y = 0, y (0) = - 6, y^{^{'}} (0) = 17

y^{^{''}} + 2 y^{^{'}} + y = 0, y (1) = 8, y^{^{'}} (0) = 6

What is the solution for y^{''}+5y^'+4y=0,y(0)=7,y^'(0)=-13 ?
The solution for y^{''}+5y^'+4y=0,y(0)=7,y^'(0)=-13 is y=5e^{-t}+2e^{-4t}