Differential Equation (Homogeneous / scale-invariant

[PhysicsMark]

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.)

$$(x+y^2)dy+ydx=0$$

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:

$$x=\alpha{x}$$

$$y=\alpha^{n}x$$

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

$$n=\frac{1}{2}$$

Based off that, I now make the following substitution:

$$y=v\sqrt{x}$$

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:

$$dy= vdx+{\sqrt{x}}dv$$

or:

$$dy= \frac{v}{2\sqrt{x}}+{\sqrt{x}dv$$

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?

 

[PhysicsMark]

So it was something else completely. It is:

$$dy = \frac{v}{2\sqrt{x}}dx+{\sqrt{x}dv$$

I think I misunderstood basic differentiation notation.