What Function Satisfies the Derivative Equation in Complex Analysis?

[Mulz]

Homework Statement

Find a function that solves the complex problem $f:C \rightarrow C$ that solves $\frac{df}{dx} = 6x + 6iy$

Relevant Equations

$\Delta u = Re(f) = \frac{df^2}{dx^2} + \frac{df^2}{dy^2}$

Mentor note: Edited to fix LaTeX problems
$f:C \rightarrow C$ that solves $\frac{df}{dx} = 6x + 6iy$

$f(x,y) = 3x^2 + 6xyi + C(y) = (3x^2 + C(y)) + i(6xy)$

$\Delta u = 0 \rightarrow 6 + C''(y) = 0 \rightarrow C(y) =5 \frac{5}{5}$

 

[Mark44]

Homework Statement:: Find a function that solves the complex problem $f:C \rightarrow C$ that solves $\frac{df}{dx} = 6x + 6iy$
Relevant Equations:: $\Delta u = Re(f) = \frac{df^2}{dx^2} + \frac{df^2}{dy^2}$

Mentor note: Edited to fix LaTeX problems
$f:C \rightarrow C$ that solves $\frac{df}{dx} = 6x + 6iy$

$f(x,y) = 3x^2 + 6xyi + C(y) = (3x^2 + C(y)) + i(6xy)$

$\Delta u = 0 \rightarrow 6 + C''(y) = 0 \rightarrow C(y) =5 \frac{5}{5}$

Note: I fixed all of your LaTeX script.

Your solution for f(x, y) looks fine to me, but I get something different for C(y).

If $f(x, y) = 3x^2 + 6xyi + C(y)$, then
$f_x = 6x + 6yi$ and $f_y = 6xi + C'(y)$

$\Delta u = 0 \Rightarrow 6 + C''(y) = 0 \Rightarrow C''(y) = -6$
Solve the last equation above to find C(y).