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

The solution for y^{''}+2ay^'+y=0 is y=c_{1}e^{(-a+sqrt(a^2-1))t}+c_{2}e^{(-a-sqrt(a^2-1))t}

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

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

y = c_{1} e^{(- a + a^{2} - 1) t} + c_{2} e^{(- a - a^{2} - 1) t}

Solution steps

y^{''} (t) + 2 a y^{'} (t) + y = 0

Solve linear ODE:

y = c_{1} e^{(- a + a^{2} - 1) t} + c_{2} e^{(- a - a^{2} - 1) t}

y = c_{1} e^{(- a + a^{2} - 1) t} + c_{2} e^{(- a - a^{2} - 1) t}

y^{^{''}} - 10 y^{^{'}} + 50 y = 0

5 \cdot x^{^{''}} + x^{^{'}} + 10 \cdot x = 0

(2 D^{5} - 7 D^{4} + 12 D^{3} + 8 D^{2}) y = 0

y^{^{''}} + 22 y^{^{'}} + 2 y = 0

y^{^{''}} + 4 y^{^{'}} + 29 y = 0, y (0) = 2, y^{^{'}} (0) = 0

What is the solution for y^{''}+2ay^'+y=0 ?
The solution for y^{''}+2ay^'+y=0 is y=c_{1}e^{(-a+sqrt(a^2-1))t}+c_{2}e^{(-a-sqrt(a^2-1))t}