Methods for coupled differential equations with elliptic functions

Question

Let's say I have a system of differential equations of the form

$$f'(x)=ag(x)+\operatorname{sn}(x;k)f(x),~g'(x)=bf(x)+\operatorname{sn}(x;k)g(x)$$

where $f$ and $g$ are some functions and $a$ and $b$ some constants, and $x$ is a real variable. What kind of methods could be used to find exact solutions to this sort of differential equation, if exact solutions do in fact exist?

Answer 1

$$f'(x)=ag(x)+\operatorname{sn}(x;k)f(x),~g'(x)=bf(x)+\operatorname{sn}(x;k)g(x)$$ $$\operatorname{sn}(x;k)=\frac{f'(x)-ag(x)}{f(x)}=\frac{g'(x)-bf(x)}{g(x)}$$ $$a-b\frac{f^2}{g^2}=\frac{f'g-g'f}{g^2}=\left(\frac{f}{g}\right)'$$ Let $\quad u=\frac{f}{g}$ $$u'=a-bu^2$$ Separable ODE : $\quad u=\sqrt{\frac{a}{b}}\tanh\left(\sqrt{ab}(x+c_1)\right)$ $$f(x)=\sqrt{\frac{a}{b}}\tanh\left(\sqrt{ab}(x+c_1)\right)g(x)$$ $$g'(x)=\sqrt{ab}\tanh\left(\sqrt{ab}(x+c_1)\right)g(x)+\operatorname{sn}(x;k)g(x)$$ $$g(x)=c_2\exp\left(\int \sqrt{ab}\tanh\left(\sqrt{ab}(x+c_1)\right)dx+\int \operatorname{sn}(x;k)dx\right)$$

Answer 2

If we let $A(x) = \begin{bmatrix}\text{sn}(x;k) & a \\ b & \text{sn}(x;k)\end{bmatrix}$ and $Y(x) = \begin{bmatrix}f(x) \\ g(x)\end{bmatrix}$, then the system can be written as $Y^\prime = AY$. For homogeneous equations of this form, we can find the fundamental matrix easily if for some $x_0\in\mathbb{R}$,

1) $B =\int_{x_0}^x A(\xi)d\xi$ exists, and

2) $AB = BA$ for all $x$

If this is the case, then the fundamental matrix is given by $\Phi(x) = \exp B$, where $\exp$ is the matrix exponential function.

For $k \neq 0$, we have $B = \begin{bmatrix}\frac{1}{k}\log\left[ \text{dn}(\xi;k) - k \ \text{cn}(\xi;k) \right] \ \big\rvert_{x_0}^x & a(x-x_0) \\ b(x-x_0) & \frac{1}{k}\log\left[ \text{dn}(\xi;k) - k \ \text{cn}(\xi;k) \right] \ \big\rvert_{x_0}^x\end{bmatrix}$.

It would be nice if there was a convenient $x_0$ such that the constant terms on the diagonal vanished, but I don't know enough about the Jacobi functions to know if one exists or what it would be. I'll leave it to you to check whether or not this commutes (again, my knowledge is limited on these functions).

If it turns out to be the case, then the fundamental matrix $\Phi$ is as defined above, and for any arbitrary constant vector $C = \begin{bmatrix}c_1 \\ c_2\end{bmatrix}$, the product $\Phi C$ is a solution to the system.