So
The whole point is to get a separable equation, right? Of course, you need to get the derivative in terms of v and x.
[EDIT]: See below for a correction.
I'm having a bit of trouble with a homogeneous ODE:
In order to solve this by substituting v = y/x, I first need to manipulate the right side to get all variables in the form of y/x. However, when I simplify it, I get:
...which leaves each fraction opposite of the other. What am I missing?
Ok, I'm with you now. This last equation looks correct to me, now that I look at it some more.
From your previous post:
Ok, here's the mistake: addition does not distribute over multiplication like that. You should do this instead:Originally Posted by Lancet
Can you continue?
Ah! I'm not sure I was ever formally taught the trick of flipping numerators/denominators on both sides - I just picked it up, thus why this aspect of it was unknown to me. Thanks!
So, taking it from here, I now end up with:
Is that the same as the book answer I referenced earlier? Or did I make another mistake? I can't quite eyeball them as being identical, but that doesn't mean anything.
The mistake occurs right here. Whenever you take the exponential of an expression such as x+c, the arbitrary constant, which was additive, becomes multiplicative:
You can redefine the arbitrary constant as to get
So, instead of
you should have gotten
Incidentally, in order to get some larger expression to show up all in an exponent (or subscript), enclose it with curly braces thus: e^{x+y}.