Jul 11, 2024

- 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.

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

**Identify M and N**: Assign M to the term in front of dx and N to the term in front of dy.**Mixed Partials Calculation**:- Calculate โM/โy and โN/โx.
- If both mixed partials are equal, the equation is exact.

**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).

**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.

**Combine Results**:- Combine the results to find the potential function F(x, y) = C.
- Check initial values, if provided, to find the particular solution.

- Given: M(x, y) and N(x, y)
- Steps:
- Identify M and N.
- Calculate โM/โy and โN/โx.
- Verify mixed partials equality.
- Choose integration variable (M w.r.t. x or N w.r.t. y).
- Integrate to find potential function.
- Find missing function by differentiation.
- Combine results (F(x, y) = C).

- 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.

- 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.

**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.

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

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

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