Definition of Differential Equations
A differential equation is an equation that involves one or more derivatives of an unknown function.
Example:
\[ \frac{dy}{dx} = 3x^2 \]
Types of Differential Equations
- Ordinary Differential Equations (ODEs): \( \frac{dy}{dx} + 2y = x^2 \)
- Partial Differential Equations (PDEs): \( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0 \)
- Linear Differential Equations: \( \frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0 \)
- Nonlinear Differential Equations: \( \frac{d^2y}{dx^2} + y^2 = 0 \)
Separation of Variables
The method of separation of variables is used to solve first-order differential equations.
A separable equation can be written as:
\[ M(x)dx = N(y)dy \]
Steps:
- Rewrite the equation in the form \( M(x)dx = N(y)dy \).
- Integrate both sides separately.
- Solve for \( y \), if possible.
Example 1: Basic Separable Equation
Given:
\[ \frac{dy}{dx} = 3x^2 \]
Separating variables:
\[ dy = 3x^2 dx \]
Integrating:
\[ y = x^3 + C \]
Example 2: Exponential Decay
Given:
\[ \frac{dy}{dx} = -ky \]
Separating variables:
\[ \frac{dy}{y} = -k dx \]
Integrating:
\[ \ln y = -kx + C \]
Taking exponent:
\[ y = C e^{-kx} \]
Video
Summary
- We introduced differential equations and their types.
- We discussed the separation of variables method.
- We solved examples using this technique.