Homogeneous Differential Equations

[nados29]

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

 

[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!

 

[saltydog]

Just to format it:

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

Hummmmm . . .

 

[Data]

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

 

[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.

 

[Hurkyl]

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.

 

[dextercioby]

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

Daniel.

 

[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.

 

[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.

 

[Sophus]

your equation is not homogeneous:
It follows from k-2m=2k-m=2k, so k=0, m=0

 

[Astronuc]

The original thread was resolved 30 months ago, or 32 months ago this month.

Halls of Ivy is correct. A new thread is appropriate for a new problem.