How can I solve this complex differential equation?

[zomfgbbq]

Homework Statement

$$\frac{dy}{dx}=e^{x+iy}$$

Homework Equations

$$e^{x+iy}=e^{x}(\cos y+i\sin y)$$

The Attempt at a Solution

$$\frac{dy}{dx}=e^{x}(\cos y+i\sin y)$$
$$\frac{dy}{\cos y+i\sin y}=e^{x}dx$$

I tried multiplying by $$\frac{\cos y-i\sin y}{\cos y-i\sin y}$$ and $$\frac{\cos y+i\sin y}{\cos y+i\sin y}$$ and tons of other bad math to try and eliminate i or get it so that I can put it with the constant. None of that led to any meaningful results. Complex analysis has never been my strong suit especially since I've never really been taught it.

 

[espen180]

Suggestion:

$$\frac{1}{\cos y + i \sin y}\cdot\frac{\cos y - i \sin y}{\cos y - i \sin y}=\frac{\cos y - i \sin y}{\cos^2 y + \sin^2 y}=\cos y - i \sin y$$

I assume the i can be taken out of the integral (after separating cos y and i sin y) since it is a constant?

 

[Hootenanny]

It would however be much simpler to note that,

$$\int\frac{dy}{e^{iy}} = \int e^x\;dx$$

$$\int e^{-iy}\;dy = \int e^x\;dx$$

$$\int \left\{\cos\left(-y\right) + i\sin\left(-y\right)\right\}\;dy = \int e^x\;dx$$

$$\int \left\{\cos\left(y\right) - i\sin\left(y\right)\right\}\;dy = \int e^x\;dx$$

which is of course the same result, but is a more straightforward approach.