Solving a diffrential equation with deriviative of more than one dependent variable

Question

How should I go about solving a differential equation of the form:

$ \frac {d}{dR}(f_1(R)g_1(R)+f_2(R) g_2 (R))=0$ where $f_1(R)$ and $f_2(R)$ are known.

I am trying to solve for $g_1(R)$ and $g_2(R)$.

Answer 1

In general in mathematics you need two equations to solve uniquely for two variables. Similarly, this one equation alone won't be enough to solve uniquely for two functions. In other words, if you want to find just one expression for $g_1$ and $g_2$ then you're going to need another equation.

That being said, integrate your equation once and you have

$$f_1(R)g_1(R) + f_2(R)g_2(R) = c$$ where $c$ is a constant. This gives you an expression relating $g_1$ and $g_2$ from which you could arrange for one of them in terms of the other.

Without further information, it's impossible to say more.

Answer 2

This is $g1 f1+g2 f2=c$ for some constant $c$ and this is one equation in two unknowns ($g1$ and $g2$).

You need more information about $f1$ and $f2$.