# Exact ODE and Finding Integrating Factors

• greentea
In summary: If the right hand side is zero, then the integrating factor is \xi_x. If the right hand side is non-zero, then the integrating factor is \xi_y.
greentea

## Homework Statement

In my ODE class, we learned how to solve first order ordinary differential equations which are not exact yet but exact after multiplying by the right integrating factor. The integrating factor we learned about take one of the five forms: f(x), f(y), f(xy), f(x/y), and f(y/x). This is to say, the integrating factors can be a function of x only, y only, x*y, x/y, and y/x.

## Homework Equations

My teacher is expecting us to solve one of these ODE among many others about 7 minutes flat. Going through all the five possible integrating factors can be very time consuming. I want to know if there are tricks to knowing which one of the five integrating factors are more likely by perhaps examining the structure of the ODE or certain characteristics that they exhibit .

If there aren't any systematically efficient way to solve ODEs that are exact by integrating factors, are there tips or tricks in general are useful in finding the right integrating factor?

Thanks!

Solving math problems is an art. You making a certain step in solving a problem may seem obvious to you but completely surprising to someone else. There is no substitute for experience, so my tip is to look at all practice problems you have and try to guess which is the correct integrating factor before attempting to solve. You should be able to see patterns.

There are some systematic ways to get integrating factors for certain types of ODE's. The ODE's that you'll encounter are probably chosen in such a way that you can almost immediately guess the integrating factor

The idea is as follows: the first order ode Q(x,y)dx - P(x,y)dy=0
admits a one-parameter group with generator $\xi\frac{\partial}{\partial x}+\eta\frac{\partial}{\partial y}$
if the function $\mu=\frac{1}{\xi Q-\eta P}$
is an integrating factor of the ODE.

So you'll need to rewrite your ODE into the Pfaffian form above, and either remember or guess the coefficients in the generator.
We know the coefficients of some ODE's, e.g.:
y'=f(x) $\xi=0,\eta=1$
y'=f(y) $\xi=1,\eta=0$
y'=f(y/x) $\xi=x,\eta=y$
y'=y/x +xf(y/x) $\xi=1,\eta=\frac{y}{x}$
y'=y/(x+f(y)) $\xi=y,\eta=0$
xy'=y+f(x) $\xi=0,\eta=x$

you can check if your generator is correct for your ODE $y'(x,y)=\omega(x,y)$ by substituting it into the equation for the linearized symmetry generator:
$\eta_x+(\eta_y-\xi_x)\omega - \xi_y\omega^2=\xi\omega_x+\eta\omega_y$

## 1. What is an exact differential equation?

An exact differential equation is a type of ordinary differential equation (ODE) that can be solved analytically without the need for any arbitrary constants. It is called "exact" because it can be written in the form of a total differential, where the variables are separated and the coefficients are functions of those variables.

## 2. How do you determine if a differential equation is exact?

To determine if a differential equation is exact, you can use the method of checking for the existence of an integrating factor. If an integrating factor can be found, then the equation is exact. Additionally, you can also check if the equation satisfies the condition of Clairaut's theorem, which states that if the partial derivatives of the equation are continuous, then it is exact.

## 3. What is an integrating factor?

An integrating factor is a function that is multiplied to both sides of an exact differential equation to make it solvable. It is usually expressed as a function of one of the variables in the equation and is used to convert the equation into a total differential form.

## 4. How do you find the integrating factor for an exact differential equation?

The general method for finding an integrating factor involves multiplying the equation by a function, taking the partial derivative with respect to one of the variables, and then equating it to the opposite of the partial derivative with respect to the other variable. This will result in an ordinary differential equation that can be solved for the integrating factor.

## 5. Why is finding an integrating factor useful in solving exact differential equations?

Integrating factors allow us to convert a non-exact differential equation into an exact one, making it easier to solve. It also helps in simplifying the process of solving differential equations, as it eliminates the need for guesswork and arbitrary constants. Additionally, integrating factors can help us find a particular solution to the equation, which can be used to find the general solution.

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