Test the following equation to show that they are scale invariant. Find their general solutions (It is not necessary to do the anti-derivative.)
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:
I then use the substitutions in the DiffEq and solve for n so that the weight of each term is equal. I find that:
Based off that, I now make the following substitution:
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.
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?