Homogeneous Differential Equations

1. Mar 21, 2005

Hi,

I need some help in finding whether this differential equation is homogeneous or not.

3 (d^2 y / dx^2) + x (dy/dx)^2 = y^2

I know that for example,

x^2 dx + xy dy = 0 is homogeneous. But how can I deal with the equation that has (d^2 y / dx^2) and (dy/dx)^2 ?

Thanks

2. Mar 21, 2005

HallsofIvy

Is your question just to determine if the equation is homogeneous or not? If so, the fact that the equation is non-linear is not relevant: yes it is homogeneous because it does not have any terms which do NOT involve y or one of its derivatives.
(That's the advantage of knowing the DEFINITION rather than just some examples.)

Of course, the fact that it is non-linear pretty much means being homogeneous doesn't make it any easier to solve!

3. Mar 21, 2005

saltydog

Just to format it:

$$3 \frac{d^2y}{dx^2}+x(\frac{dy}{dx})^2-y^2=0$$

Hummmmm . . .

4. Mar 21, 2005

Data

Indeed, homogeneous but additionally nonlinear. Quite analytically insoluble, though.

5. Mar 21, 2005

Crosson

The simplist way to answer the question of homogeneity is to ask:

Is Y(x) = 0 a solution?

If the answer is yes, then the equation is homogeneous.

6. Mar 21, 2005

Hurkyl

Staff Emeritus
Hrm, does it really make sense to ask if a nonlinear DE is homogenous? I don't have a general definition handy, and Mathworld only defines homogeneity for linear differential equations.

7. Mar 21, 2005

dextercioby

Probably Mathworld gives attempts to solve it,too...Is a nonlinear algebraic system either homogenous or nonhomogenous...?

Daniel.

8. Sep 3, 2007

cheeky_99

Hey i need some help finding the general solution of

ydy= (-x+ √(x^2 + y^2))dx

by using the substitution y= vx and then the substitution u^2= 1 + v^2

It would be great if someone could help.

9. Sep 3, 2007

HallsofIvy

Do not, do not, do not "hijack" someone else's thread for a new question. It's very easy to start a thread of your own!

In fact, I'm going to do that for you.

10. Nov 9, 2007