Differential Equations
Study of equations involving derivatives and their applications.
Basic Concepts
First-Order Differential Equations
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
- Rearrange the equation.
- Find an integrating factor (IF): \( IF = e^{\int P(x)dx} \)
- Multiply through by IF and integrate both sides.
- Solve for \( y \).
Separable Equations
These can be written as \( \frac{dy}{dx} = f(x)g(y) \) and solved by separating variables and integrating.
Examples
\( \frac{dy}{dx} = 3y \) models exponential growth.
\( \frac{dy}{dx} = x \) solves to \( y = \frac{1}{2}x^2 + C \).
In a Nutshell
First-order equations involve the first derivative and are often solvable by simple methods.