We are given the equation:

$\cos(x + y) \, dy = dx$

This is a first-order differential equation. To solve it, we can rearrange the terms and then integrate both sides.

Step 1: Rearrange the equation

We can rewrite the equation as:

$\frac{dy}{dx} = \frac{1}{\cos(x + y)}$

This is a separable differential equation, meaning we can separate the variables $x$ and $y$ on each side.

Step 2: Separate the variables

Let's move all terms involving $y$ to one side and those involving $x$ to the other side:

$\cos(x + y) \, dy = dx$

This is already separated. However, it is easier to solve if we make a substitution.

Step 3: Substitution

Let’s introduce a new variable $z = x + y$, so that $dz = dx + dy$.

Rewriting the equation in terms of $z$, we get an integral on both sides, which we can solve.

Would you like to go through the details step by step?

Here are some further questions related to this topic:

1. How do you solve separable differential equations?
2. What is the process of substitution in solving differential equations?
3. Can this equation be solved using a different method?
4. What is the geometrical interpretation of this equation?
5. What are the initial conditions needed to find the particular solution?

Tip: Always check if a differential equation is separable or can be simplified using substitution—it can make solving much easier.