Exact Differential Equations Lecture Notes

Introduction

• Focus: Solving exact differential equations and finding potential functions.
• Previous video: Explained what exact differential equations are, their origin, and solving method.
• Plan: Practice solving several examples, discuss integrating factors, and work with initial values.

Key Concepts

1. Exact Differential Equation: A form of a total differential or exact differential if the mixed partials are equal (continuous and differentiable on some region).
2. Potential Function: The function F(x, y) that gives the exact differential equation when its total differential is taken.
3. Mixed Partials Test: Check if the mixed partial derivatives of M (∂F/∂x) and N (∂F/∂y) are equal.

Steps to Solve Exact Differential Equations

1. Identify M and N: Assign M to the term in front of dx and N to the term in front of dy.
2. Mixed Partials Calculation:
   • Calculate ∂M/∂y and ∂N/∂x.
   • If both mixed partials are equal, the equation is exact.
3. Integral Calculation:
   • Choose to integrate either M with respect to x or N with respect to y.
   • If chosen integral, derive the partial derivative of the potential function and set it equal to the corresponding function (other partial derivative).
4. Find Missing Function:
   • After integration, include a function of the other variable to represent unknown terms.
   • Use differentiation and setting equal to other partial derivative to solve for the unknown function.
5. Combine Results:
   • Combine the results to find the potential function F(x, y) = C.
   • Check initial values, if provided, to find the particular solution.

Example Walkthroughs

Example 1: Simple Test and Solution

• Given: M(x, y) and N(x, y)
• Steps:
  1. Identify M and N.
  2. Calculate ∂M/∂y and ∂N/∂x.
  3. Verify mixed partials equality.
  4. Choose integration variable (M w.r.t. x or N w.r.t. y).
  5. Integrate to find potential function.
  6. Find missing function by differentiation.
  7. Combine results (F(x, y) = C).

Example 2: Complex Partial Integration

• Discusses importance of integrating the right function based on simplicity.
• Demonstrates frequent use of integration by parts and substitutions.
• Process repeated for M and N integrations.

Example 3: Including Initial Values

• Initial values provided to find a particular solution:
  • Once the potential function is found, substitute initial values for both x and y.
  • Solve for the constant C.

Important Observations

• Integral Choice: Choose the function (M or N) with simpler integration.
• Boundary Conditions: Include boundary conditions early to avoid numerical errors.
• Function Notation: Clearly distinguish functions of x (G(x)) and functions of y (H(y)).
• Constants: Understand that a plus constant is implicit and sometimes needs to be handled separately.

Tips for Solving Exact Differential Equations

1. Always verify mixed partials equality before proceeding.
2. Integrate the easier term first to simplify computations.
3. Include arbitrary functions of the other variable during integration.
4. Use initial values to find specific solutions when available.
5. Practice multiple examples to gain confidence and familiarity with various integral forms.

Next Steps

• Learn about integrating factors in the next video to handle non-exact equations.
• Practice additional examples to reinforce understanding of exact differential equations technique.

Conclusion

• Reinforcement of the solving steps and importance of verifying conditions (mixed partials)
• Message of encouragement and readiness for the next lesson.