What is the solution for y^{''}+5/2 y^'+y=0 ?

The solution for y^{''}+5/2 y^'+y=0 is y=c_{1}e^{-t/2}+c_{2}e^{-2t}

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y^{''}+5/2 y^'+y=0

y^{^{''}} + \frac{5}{2} y^{^{'}} + y = 0

y = c_{1} e^{- \frac{t}{2}} + c_{2} e^{- 2 t}

Solution steps

y^{''} (t) + \frac{5}{2} y^{'} (t) + y = 0

Solve linear ODE:

y = c_{1} e^{- \frac{t}{2}} + c_{2} e^{- 2 t}

y = c_{1} e^{- \frac{t}{2}} + c_{2} e^{- 2 t}

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

\frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} - 4 y = 0

y^{^{''}} + (e + 1) y^{^{'}} + ey = 0

3 y^{^{''}} - y^{^{'}} = 5 y

y^{^{''}} - y^{^{'}} - 42 y = 0, y (0) = 1, y^{^{'}} (0) = - 1

What is the solution for y^{''}+5/2 y^'+y=0 ?
The solution for y^{''}+5/2 y^'+y=0 is y=c_{1}e^{-t/2}+c_{2}e^{-2t}