Second Order Differential Equation

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$\ddot{x}+\frac{2\sqrt{15}}{3}\dot{x}+\frac{5}{3}x=0$, general form $e^{-wt}(At+B)$

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You have a linear diff eq with constant coefficients.

The characteristic equation is $\lambda^2 +\frac {2\sqrt 15}{3} \lambda + \frac 53 = 0$

find the roots of the characteristic equation.

There are three possibilities:

you have two real roots.

You have two complex roots.

You have one real root of multiplicity 2.

What have you got? What is the solution when the characteristic equation has that form?