How to divide solutions of system of ODE

Question

If I have the following system of ODEs: \begin{align} F'_1=aF_1+bF_2 \nonumber\\ F'_2=-bF_1+cF_2 \nonumber \end{align}for which I know the solutions of $F_1$ and $F_2$ as sums of exponential of eigenvalues. My question is can I find an ODE for $z=\dfrac{F_1}{F_2}$ . I want this so that I can find the following integral $$\int_{t_0}^t\dfrac{F_1(s)}{F_2(s)}ds$$ explicitly.

Answer 1

Assuming $F_2 \neq 0$, you can use the quotient rule to get the following (non-linear) ODE for $F_1/F_2$:

$$\left(\dfrac{F_1}{F_2}\right)' = \dfrac{F_1'F_2 - F_1F_2'}{F_2^2}=\dfrac{(aF_1 + bF_2)F_2 - F_1(-bF_1 + cF_2)}{F_2^2}$$ $$ =\dfrac{bF_1^2 + (a - c)F_1F_2 + bF_2^2}{F_2^2} $$ $$ = b\left(\dfrac{F_1}{F_2}\right)^2 + (a-c)\dfrac{F_1}{F_2} + b $$

If you just want to find the value of that integral, you can use the solutions you know from the system to (maybe) calculate it directly. E.g. the integral will be something like

$$ \int_{t_0}^t \frac{c_1 a e^{\lambda_1 t} + c_2 b e^{\lambda_2 t}}{c_1 c e^{\lambda_1 t} + c_2 d e^{\lambda_2 t}} \,dt $$

for some (different) constants $c_1,c_2,a,b,c,d$. You may not be able to explicitly find an (elementary) anti-derivative, but you will still be able to evaluate it numerically and perhaps study other aspects of its behavior.