$f=u+iv$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$? $f(z)=u(x,y)+iv(x,y)$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?
I tried to interprete $xu+yv$ as some part of a new function, for example, as the real part of $\overline{z}f$，but this function is no more holomorphic, so I don't know how to continue. (Maybe $\dfrac{\partial}{\partial\overline{z}}（\overline{z}f)=1$? But how to make use of it?)
 A: What you need to notice is that
$$ \frac{f(z)}{z} = \frac{u+iv}{x+iy} = \frac{(u+iv)(x-iy)}{x^2+y^2}, $$
of which the real part is
$$ \frac{xu+yv}{x^2+y^2}, $$
which is equal to $e^x \cos{y}$ according to the problem. Now find the harmonic conjugate of $e^x \cos{y}$, which is well-known to be $e^x \sin{y}$ for obvious Euler's formula-related reasons, and rewrite the equation to include the imaginary part:
$$ \frac{f(z)}{z} = e^x(\cos{y}+i\sin{y}) = e^{x+iy}=e^z. $$
A: We have $xu+yv=(x^2+y^2)e^x\cos y$.  For notational efficiency, let $G(x,y) = (x^2+y^2)e^x\cos y$.
Here is a procedure for finding 

Step 1:  
Take the partial derivative with respect to $x$ and multiply by $x$ to find that 
$$x^2u_x+xu+xyv_x=xG_x$$
where the subscript $x$ means the first partial with respect to $x$.

Step 2:  
Take the partial derivative with respect to $y$ and multiply by $y$ to find that 
$$y^2v_y+yv+xyu_y=yG_y$$
where the subscript $y$ means the first partial with respect to $y$.

Step 3:
Now, add these last two equations together and apply the Cauchy-Riemann equations ($u_x=v_y$ and $u_y=-v_x$).  This reveals that 
$$(x^2+y^2)u_x+G=xG_x+yG_y$$
which implies that 
$$u_x=v_y=\frac{xG_x+yG_y-G}{x^2+y^2}$$

Step 4:
Integrate with respect to $x$ to find $u$ within and integration constant and integrate with respect to $y$ to find $v$ within and integration constant.  

Step 5:
Enforce $xu+yv=G$ for all $x$ and $y$ to find the integration constants.  Then $f=u+iv$.
