Exact Diff Eq & Integrating Factors

Lecture Notes on Exact Differential Equations and Integrating Factors

Introduction to Exact Differential Equations

• Definition: A differential equation of the form
  $M(x,y)dx + N(x,y)dy = 0$ is exact if there exists a function ( F(x,y) ) such that
  ( \frac{\partial F}{\partial x} = M ) and ( \frac{\partial F}{\partial y} = N ).

• The method of solving such equations typically involves finding this function ( F ), which is often done through integration of the components in relation to their respective variables.

Non-Exact Equations

• Sometimes, an equation of the form
  $M(x,y)dx + N(x,y)dy = 0$ is not exact. In these cases, an integrating factor may be necessary to transform the equation into an exact form.

Definition of Integrating Factor

• An integrating factor for a non-exact equation is defined as a function ( \mu(x,y) ) such that when multiplied by the original equation, it produces an exact equation.

Finding Integrating Factors

• If the equation is not exact, we may look for possible forms of an integrating factor. Here are two typical cases:

  • Option #1: Assume that ( \mu = \mu(x) ) (depends only on ( x )). In this case, the following condition must hold:
    $\frac{\partial (\mu M)}{\partial y} = \frac{\partial (\mu N)}{\partial x}$

  • Option #2: Assume that ( \mu = \mu(y) ) (depends only on ( y )). Here too, the same condition is required to hold.

Steps for Finding Integrating Factors

1. Check for Option #1: ( \mu = \mu(x) )

   • If ( \mu M ) with respect to ( y ) gives equal derivatives when compared with ( \mu N ) with respect to ( x ), then this function can be an integrating factor.

   • If true, you can proceed to integrate and find solutions.

2. Check for Option #2: ( \mu = \mu(y) )

   • The same relationships apply, checking the derivatives in relation to each variable to see if they yield equivalence when considered with the appropriate functions of ( M ) and ( N ).

Example Problem

• Consider the differential equation:
  $M = xy, N = 2x^2 + 3y^2 - 20$

• Here,

  • ( M ) and ( N ) are not equal, confirming it as a non-exact equation.

• To solve the system, check the options:

  • Option #1: Check if
    $2N - M = \text{is a function of } x?$
    If not,

  • Option #2: Check if
    $N = N(y) \text{is a function of } y.$

• If either yields valid conditions, use that integrating factor to solve the original equation.

Final Notes and Insights

• After integrating and solving, it’s important to reflect on whether your solutions have any relationships with the original problems or setups.

• Evaluation/checking against different formats of desired outcomes such as ( F(x,y) = c ) or adjustments to forms can be essential for ensuring correctness.

Summary

• Exact and non-exact equations require distinct approaches for solutions. The concept of integrating factors serves as a tool for transforming non-exact equations into a solvable format. Each function's independence plays a crucial role in determining the paths we take to integrate and find those solutions efficiently.