Finding Second Order Linear Equation

Question

I need to write down the second order linear equation which has v(t) as its solution. $A>1$ and$g <\dfrac{ 4(A-1)}{A^2} $ . $v(t)$ is an underdamped harmonic oscillation.

Given: $u' = -g(Au+v) \\ v' = (A-1)u $

I know how to make a second order equation into a first order one, but no clue how to do the opposite.

Thanks. Any help is appreciated.

Answer 1

hint :$$\\\begin{cases}u' = -g(Au+v)\\v' = (A-1)u\end{cases}\\ \to \begin{cases}u' = -g(Au+v) &\to u''=-g(Au'+v') \\v' = (A-1)u\end{cases}\\$$now put $v'$ into first one $$ u''=-g(Au'+\color{red} {v'}) \to u''=-g(Au'+\color{red} {(A-1)u}) $$