Solve e^ix=cosx+ydx

$$e ^ { i x } = \cos x + y d x$$

Solve for d

$$\left\{\begin{matrix}d=\frac{i\sin(x)}{xy}\text{, }&x\neq 0\text{ and }y\neq 0\\d\in \mathrm{C}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\cos(x)+ydx=e^{ix}$$

Subtract \cos(x) from both sides.

$$ydx=e^{ix}-\cos(x)$$

The equation is in standard form.

$$xyd=e^{ix}-\cos(x)$$

Divide both sides by yx.

$$\frac{xyd}{xy}=\frac{i\sin(x)}{xy}$$

Dividing by yx undoes the multiplication by yx.

$$d=\frac{i\sin(x)}{xy}$$