I was wondering if there is a solution for the following problem. Assume I have a one-dimensional system of the following form
$$u(x) = v(x) f(x) + w(x) \frac{\partial f(x)}{\partial x} $$
where $u(x)$, $v(x)$, and $w(x)$ are known $\mathbb{R}\rightarrow\mathbb{R}$ functions. It is my goal to find one (or multiple) $f(x)$ which fulfill this equation.
Now my questions:
- Assuming that a solution exists, do you know of a method that would help to me identify $f(x)$?
- If yes: I am aware that depending on $u(x)$, $v(x)$, and $w(x)$, there might not be a solution. Is there a way to recognize whether this might be the case?
I would also appreciate any partial answers of pointers in promising directions!
An (optional) simplification, if it helps:
We can simplify the problem above by assuming that we want to solve the equation above on a discretized line, that is to say, $u(x)$, $v(x)$, $w(x)$, and $f(x)$ take on scalar real values on $n$ nodes $x_1,...,x_n$ on a line, and $\frac{\partial f(x)}{\partial x}$ can be approximated with finite differences. In that case, can the two questions above be answered?
This is a first-order differential equation. If you assume that $w(x)>0$ for all $x\in\mathbb{R}$, then this expression can be rewritten as
$$f'(x)=-\dfrac{v(x)}{w(x)}f(x)+\dfrac{u(x)}{w(x)}.$$
Let's say $f(x_0)=f_0$ for some $x_0\in\mathbb{R}$. If we define $$\Phi(x,y)=\exp\left(-\int_{y}^x\dfrac{v(\theta)}{w(\theta)}d\theta\right),\ x\ge y$$
then we have that
$$f(x)=\Phi(x,x_0)f_0+\int_{x_0}^x\Phi(x,s)\dfrac{u(s)}{w(s)}ds.$$