I want to find a solution of a system of 1st order ODEs.
$Z_t=2(P^2+Q^2)$
$P_t=2PZ$
$Q_t=2QZ$
To solve this, I tried to differentiate the first equation and get
$Z_{tt}=4PP_t+4QQ_t=8Z(P^2+Q^2)=4ZZ_t$
which is a second order nonlinear ODE for Z.
If I find the solution of Z in general form, it is easy to find what P and Q is. However, It is hard to solve on Z and I failed.
Is there anyone who can solve that ODE for Z or can solve the system of ODEs on Z,P,Q in other ways?
Please help me. Thank you for your any comments or advices.
Your approach is correct so far. Therefore $$ Z'= 2Z^2 + C, $$ where the constant $C$ can be determined from $ C = Z'(0) - 2Z(0)^2 = P(0)^2 + Q(0)^2 - 2 Z(0)^2 \, . $ This can now be solved via separation of variables.