First-Order Equations

These equations involve only the first derivative of a function. They are the simplest kind and a great starting point for learning.

General Form

\( \frac{dy}{dx} + P(x)y = Q(x) \)

Here, \( y \) is the unknown function, and \( P(x) \) and \( Q(x) \) are known functions.

Solving First-Order Linear Equations

1. Rearrange the equation.
2. Find an integrating factor (IF): \( IF = e^{\int P(x)dx} \)
3. Multiply through by IF and integrate both sides.
4. Solve for \( y \).

Separable Equations

These can be written as \( \frac{dy}{dx} = f(x)g(y) \) and solved by separating variables and integrating.