About solving a first order differential equation

Question

I have the following question which I cannot find a solution

Find a function $g$ verifying:

$$(x-f(x))g′(x)=g(x)$$

where $f$ is a given function defined on a closed interval $[a,b]$.

Answer 1

$$\int\frac{dg}{g}=\int\frac{dx}{x-f}$$ $$\ln|g|=\int\frac{dx}{x-f}$$ $$g=e^{\int\frac{dx}{x-f}}$$ Badda boom badda bing you have your solution. Keep in mind that $\int\frac{dx}{x-f}$ may not have an elementary form depending on your choice of $f$.