Looking for the name of a class of ODE:

[Monochrome]

Homework Statement

$$M(x,y)y^{'}(x) + N(x,y) = 0$$
There exists:
$$\phi(x,y)$$
Such that
$$\frac{\partial\phi(x,y)}{\partial x}=N(x,y)$$

$$\frac{\partial\phi(x,y)}{\partial y}=M(x,y)$$

I'm not looking for a solution to anything particular to this but I can't find the type in my notes and I can't google it unless I know the name.

 

[HallsofIvy]

That is an exact equation since
[tex]d\phi= M(x,y)dy+ NIx,y)dx[/itex]
is an exact differential.

Of course, since the differential equation says $d\phi= 0$, $\phi(x,y)= 0$ is the general solution.

 

[HallsofIvy]

By the way, in physics, such a differential would correspond to a "conservative force field" and the function $\phi$ would be the "potential function".

 

[Monochrome]

Thanks.
Just a question: The I after the N is a typo yes? And how did you get $d\phi= 0$?

 

[HallsofIvy]

Yex, that was a typo- my finger was aiming at "("!

Since $\phi$ is a function of both x and y, $d\phi /dx$ would make no sense. By the chain rule, if x and y are functions of some third variable, t,
$$\frac{d\phi}{dt}= \frac{\partial \phi}{\partial x}\frac{dx}{dt}+ \frac{\partial \phi}{\partial y}\frac{dy}{dt}$$
or, in differential notation,
$$d\phi= \frac{\partial \phi}{\partial x}dx+ \frac{\partial \phi}{\partial y}dy$$

 

[Monochrome]

Yex, that was a typo-

:rofl:

Thanks I forgot that this was dealing with partials. It makes sense now.