Indefinite integrals are really useful for solving differential equations. Let’s break it down step by step:

1. Finding General Solutions: When you have a first-order differential equation, like $\frac{dy}{dx} = f(x)$, you can integrate both sides. This means you find $y$ by doing $y = \int f(x) \, dx + C$. The $C$ here is a constant. It holds all possible answers!

2. Initial Conditions: If you have a specific point, like $y(x_0) = y_0$, you can use this in your general solution to figure out $C$. This helps you find a unique solution that works for your specific problem.

3. Higher Orders: For second-order equations, which look like $\frac{d^2y}{dx^2} = f(x)$, you need to integrate twice. Each time you do this, you add a new constant.

In summary, using indefinite integrals is like having a map to guide you when solving differential equations!