Differential Equations : Integrating factors

1. Sep 18, 2011

pooja mehta

If u(x,y) and v(x,y) are two integrating factors of a diff eqn M(x,y)dx + N(x,y)dy,
u/v is not a constant, then
u(x,y) = cv(x,y) is a solution to the differential eqn, for every constant c.

2. Sep 18, 2011

Dickfore

What does an integrating factor mean?

3. Sep 19, 2011

wisvuze

an "integrating factor" is a function F ( for example ) so that when you multiply the differential equation out by F, the result is easier to integrate ( i.e. with a product rule or something )

4. Sep 19, 2011

Dickfore

What is the strict condition that has to be satisfied?

5. Sep 19, 2011

wisvuze

the integrating factors are more of a technique than something that is explicitly defined; certain types of equations will have different conditions for the integrating factors ( as the thing the OP wants to show is a condition about the integrating factors ).. or for an equation of the form y ' + p( t )y = g( t ), an integrating factor can be of the form e^integral ( p( t ) ) [ when you multiply the equation out by this, you can easily integrate because the expression will be in the "product rule form" ]

6. Sep 19, 2011

Dickfore

But, we are considering a different kind of equation here, namely the so called exact differential equation:

$$M(x, y) \, dx + N(x, y) \, dy = 0$$

What does it mean in this context?

7. Sep 20, 2011

HallsofIvy

Theoretically, every first order differential equation has an "integrating factor". It is only for linear equations that it is possible to give a general formula for that integrating factor.

Saying that u(x,y) is an integrating factor for the diffrerential equation M(x,y)dx+ N(x,y)dy= 0 means that u(x,y)M(x,y)dx+ u(x,y)N(x,y)dy= d(f(x,y)) for some differentiable function f(x,y). That, of course, means that u(x,y)M(x,y)dx+ u(x,y)N(x,y)dy is "exact". In particular that
$$\frac{\partial f}{\partial x}= u(x,y)M(x,y)$$
and that
$$\frac{\partial f}{\partial y}= u(x,y)N(x,y)$$

8. Sep 20, 2011

Dickfore

So, if those are the first partial derivatives, what are the second mixed derivatives?