To solve first-order differential equations, we have several methods we can use. Each one works best for different kinds of problems. Here are some important techniques:

1. Separation of Variables: This method works well when we can keep the variables apart. We can change the equation to look like $f(y) dy = g(x) dx$. For example, if we start with the equation $dy/dx = xy$, we can rearrange it to make it easier to work with both sides.

2. Integrating Factors: This technique is handy for linear equations that look like $dy/dx + P(x)y = Q(x)$. If we multiply the whole equation by an integrating factor, we can make it simpler and find the solution.

3. Homogeneous Equations: If we can write the equation in the form $dy/dx = f(y/x)$, we can use a substitution. We set $v = y/x$ to make the equation easier to solve.

4. Exact Equations: An equation is exact when $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. If this is the case, we can find a potential function that will help us solve the equation.

Each of these methods gives us a way to find solutions. They help us understand how the functions related to these equations behave.