Differential Equations - Lecture 3: Exact Differential Equations

Introduction

Previous Lectures
- First-order differential equations
  - Discussed order and degree
  - Methods: Variable separable, Homogeneous differential equation
- Linear differential equations
  - Definition and solution techniques

Definition: A differential equation is exact if it can be written in the form (M(x, y)dx + N(x, y)dy = 0) and the partial derivatives are equal (\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}).
Identifying Exact Equations:
1. Check if variable separable method is applicable.
2. If not, check for reducibility to variable separable method.
3. Check for homogeneous function.
4. Check if it's linear in x or y.
5. Calculate values to see if (\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}).
Solving Exact Equations:
- Calculate the integrals involved.
- Short-cut trick: Directly integrate M with y as constant and N with x-excluded terms.
- Method: Partial differentiation to verify equality, then integrate.
Example Questions:
1. Verify exactness through differentiation.
2. Apply the solving method as shown step-by-step.

When Not Exact:
- Check if it can be made exact using integrating factors.
- Three approaches:
  1. Homogeneous differential equations
  2. Common factors (x) or (y)
  3. Direct method
Homogeneous Differential Equations:
- Verify homogeneity (all terms same degree).
- Calculate integrating factor if non-exact.
- Multiply equation by integrating factor and check exactness.
- Example: Calculating integrating factor and reducing to exact form.
Common Factor Method:
- Case 1: y common ( Multiply m by x and n by y for integrating factor)
- Case 2: x common ( Example calculations and solutions)
Direct Method:
- Smaller term in denominator.
- Calculate integrating factor and solve.
- Examples: Step-by-step simplification and integration.