I have the following O.D.E.: $$\frac{\Bbb dy}{\Bbb dx} = 2u.$$ Above, $u$ is a function of $x$ and $y$. My gut tells me that this equation is separable. In this way, do I have a “partial” integral? I have no idea how to solve this as $u$ is unknown and is also a function of $x$.
Thanks for the help!
$$\frac{\Bbb dy}{\Bbb dx}=2u(x,y)=:F(x,y)\tag{1}$$ The above expression is the general form of a first-order O.D.E. There is no general solution to an arbitrary eqn. such as $(1)$ without given $F$ - however, there are theorems predicting the existence of solutions. Perhaps that’s what you’re looking for. Let me write one:
Picard’s Theorem: Let $R$ be a rectangle, and suppose that $F(x,y)$ and $\partial F/\partial y$ are continuous on $R$. Also, suppose that $(x_0,y_0)$ is an interior point of $R$. Then, $(1)$ with the initial value $y(x_0)=y_0$ has a unique solution $y=y(x)$ for $x$ in some open interval $\mathcal{O}$ such that $x_0\in\mathcal{O}$.
With the conditions of the above theorem, we see that any solution to $(1)$ with the initial value $y(x_0)=y_0$ must satisfy $$y(x)=y_0+\int^x_{x_0}F\big(t,y(t)\big)\,\Bbb dt.$$ This, I guess, is the “general” solution of $(1)$ with the initial value $y(x_0)=y_0$.
Hope this helps.