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Differential Equation (Homogeneous / scale-invariant

  • #1

Homework Statement


Test the following equation to show that they are scale invariant. Find their general solutions (It is not necessary to do the anti-derivative.)

[tex](x+y^2)dy+ydx=0[/tex]

I believe what my tutorial wants me to do is to check for homogeneity. I'm not sure though. This is not a Differential equations class, it is a math methods in physics class. The tutorial we use titles this section "Scale-invariant first-order differential equations".


The Attempt at a Solution



First I make the following substitutions:

[tex] x=\alpha{x}[/tex]

[tex] y=\alpha^{n}x[/tex]

I then use the substitutions in the DiffEq and solve for n so that the weight of each term is equal. I find that:

[tex]n=\frac{1}{2}[/tex]

Based off that, I now make the following substitution:

[tex] y=v\sqrt{x}[/tex]

From there things begin to get messy. I am unsure about 1 step so far: I am not sure what dy equates to. In the case that n=1, dy = vdx+xdv. I am not sure how to get dy using n=1/2.

Is it:

[tex]dy= vdx+{\sqrt{x}}dv[/tex]

or:

[tex]dy= \frac{v}{2\sqrt{x}}+{\sqrt{x}dv[/tex]

Or something else completely?

I guess I am asking what is dx in the equations above? Is it the derivative of x^1/2 or is it just dx?
 

Answers and Replies

  • #2
So it was something else completely. It is:

[tex] dy = \frac{v}{2\sqrt{x}}dx+{\sqrt{x}dv[/tex]

I think I misunderstood basic differentiation notation.
 

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