I am doing numerical analysis where we work with differential equations but I have never had any classes on differential equations. It seems you can get by in an introductory numerical analysis course with just knowing what a differential equation is an how the initial value problem solving process works. So I know a few process for solving them numerically but I have never learned how to solve them mathematically.
Anyway, I would like to know how to 'create' differential equation for testing my numerical approximation algorithms against, but as I have never learned how to deal with them mathematically I don't know the process for creating them. Can someone show me a step by step process for creating simple differential equations?
If I have $y = 5x^2 +c$, $c$ being some constant
I can make a differential equation of the form -
$\frac{dy}{dx} = 10x$
But I want to have differential equations where the $\frac{dy}{dx}$ is dependent on not just $x$, but on $x$ and $y$.
What is the process for doing this?
Edit
To clarify, I am looking to create simple equations of the form $\frac{dy}{dx} = f(x,y)$, $y(0) = y_0$ with solutions $y$ that are also relatively simple. This is so I can then write out the first few iterations of RK methods by hand to see how it all works.