Those are two different ways of writing one thing.
The y' notation is sometimes called Newton notation (although he used a dot instead of a prime), and the other is called Leibniz notation.
#4
PhysicsMark
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I think I may have answered my own question with this next one, but I would like to get a confirmation.
I asked the question above because in my math methods in physics class, the tutorials we use and the professor often algebraically manipulate the dy's and dx's in equations involving y'.
I am reviewing scale invariance and its applications to FODE's.
After proving the following equation is scale invariant, I am to solve for the general solution.
[tex]y'+\frac{y^{2}}{x^{2}}=2[/tex]
After I have shown it is scale invariant, I use the substitution y=vx to obtain:
[tex]v+x\frac{dv}{dx}+v^2=2[/tex]
In order to get to the point above, I note that:
[tex]dy=vdx+xdv[/tex]
Here is where my question comes in. I originally looked at this problem by stating:
As far as I know, v is a number not a function of x. I believe I understand that this method is used to make an inhomogeneous equation separable. How do I avoid making the wrong simplification?
I think the answer is that I need to recognize that I must have a "dv" and a "dx" in order to solve the equation in a separable manner, otherwise I would have had:
[tex]dv+v^2=2[/tex]
Does that make any sense at all? Thanks for replying.
#5
PhysicsMark
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Haha...nevermind. It is the same thing, just not carried out all the way through.