Problem: Find out the solution of the Cauchy problem for the first order partial differential equation \[ x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = z, \] on $D= \left\{ (x,y,z):x^{2} +y^{2} \neq 0,~z>0 \right\} $. The initial condition is $x^{2} +y^{2} =1,~z=1$.
Solution: I encourage you to attempt to solve the problem today. The solution will be provided tomorrow. This will give you the opportunity to test your understanding of the problem and to improve your skills in solving similar problems in the future.