Homogeneous vs. Exact (w/ I.F.)

[NSX]

Hello!

 

My D.E. exam is tomorrow, and I was wondering something (but don't have the time to try it myself).

 

Say you have a first order diff. eq.:

 

e.g.

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

 

Now, say that one method to solve it in y is to use the homogenous method, then seperating (i.e. y = v(x)*x, dy/dx = ...)

 

Could an integrating factor solve this equation too (in general)?

 

More specifically, an integrating factor w/rt to only one variable, either some u(x) or u(y) [such that Mu dx + Nu dy = 0 is an exact D.E.].

 

Thanks!

[bloodhound]

M(x,y)dx +N(x,y)dy=0? i dont get the equation. are you multiplying by differentials dx and dy, or are you trying to say derivative of M wrt x derivative of N wrt y.

[NSX]

M(x,y)dx +N(x,y)dy=0? i dont get the equation. are you multiplying by differentials dx and dy, or are you trying to say derivative of M wrt x derivative of N wrt y.

 

Well, I think more precisely, it should be the letter "delta" where "d" goes.

 

The dx & dy are the partials of some function, F, which has x and y variables in it.

 

i.e. dF(x,y) = M(x,y)dx + N(x,y)dy

 

But I think your former statement is the same too.

 

[edit]

 

Like the D.E. described here: http://www.efunda.com/math/ode/ode1_exact.cfm

Homogeneous

http://www.tau.ac.il/~levant/ode/solution_4.pdf

 

[edit - 2]

 

n/m

 

I remember now that first order D.E. are only homogeneous if they have this property:

 

f(tx,ty) = t*f(x,y)