What is the solution for (d^2x)/(dt^2)+5(dx)/(dt)+4x=0,x(0)=1 ?

The solution for (d^2x)/(dt^2)+5(dx)/(dt)+4x=0,x(0)=1 is x=(1-c_{2})e^{-t}+c_{2}e^{-4t}

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(d^2x)/(dt^2)+5(dx)/(dt)+4x=0,x(0)=1

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

x = (1 - c_{2}) e^{- t} + c_{2} e^{- 4 t}

Solution steps

\frac{d ^{2} x}{d t ^{2}} + 5 \frac{d x}{d t} + 4 x = 0

Solve linear ODE:

x = (1 - c_{2}) e^{- t} + c_{2} e^{- 4 t}

x = (1 - c_{2}) e^{- t} + c_{2} e^{- 4 t}

x^{^{''}} + a \cdot x = 0

y^{^{''}} - 3 y^{^{'}} + 2 y = 0, y (1) = 0, y^{^{'}} (1) = 1

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

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

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

What is the solution for (d^2x)/(dt^2)+5(dx)/(dt)+4x=0,x(0)=1 ?
The solution for (d^2x)/(dt^2)+5(dx)/(dt)+4x=0,x(0)=1 is x=(1-c_{2})e^{-t}+c_{2}e^{-4t}