Find an expression for $f(z)$ using Cauchy-Riemann Equations:

Question

Suppose that $f: \mathbb{C} \to \mathbb{C}$ with $f(z)=u(x,y) + iv(x,y)$ is differentiable and that

$3u(x,y) + v(x,y) = 7$, $\forall z=x+iy \in \mathbb{C}$.

Find an expression for $f(z)$.

I've been told that I need to solve this using the Cauchy-Riemann equations(CRE) by finding a system of the partial derivatives from an expression like $f=u+i(7-3u)$. I understand how CRE work but can't work out how to find the solution in this case.

Answer 1

From $f=u+i(7-3u)$ we get by the CRE:

$u_x=-3u_y$ and $u_y=3u_x.$

Can you proceed ?