Find the analytic function $()=+$ such that $−=(+)(−)$

Question

Find the analytic function $()=+$ such that $−=(+)(−)$.

I got $u_x=2x+v_x$ and $v_y=u_y+2y$.

To be analytic $u_x=v_y$ and $u_y=-v_x$.

Iam stuck after the above step.Tried summing both the equation ,still not sure about it

Answer 1

We are given $ u = x^2 - y^2 -v$. For $f(z) = f(x+iy) $ to be analytic it has to satisfy the Cauchy-Riemann equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} $ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $.

Substitute our expression for u into the first relation to find: $ 2x= \frac{\partial v}{\partial y} + \frac{\partial v}{\partial x}$. This is a PDE which should be solvable for $v(x,y)$ whereafter $u(x,y)$ can be found.