What is the solution for x^{''}+2abx^'+b^2x=0 ?

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

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x^{''}+2abx^'+b^2x=0

x^{^{''}} + 2 ab x^{^{'}} + b^{2} x = 0

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

Solution steps

x^{''} (t) + 2 ab x^{'} (t) + b^{2} x = 0

Solve linear ODE:

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

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

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