The discussion revolves around solving a homogeneous differential equation using the substitution \( v = \frac{y}{x} \). The original poster presents the equation \( \frac{dy}{dx} = \frac{3y^2 - x^2}{2xy} \) and attempts to manipulate it into a more workable form.
Some participants have provided guidance on how to proceed with the substitution and have pointed out errors in the original poster's calculations. The conversation indicates progress in understanding the problem, though there is no explicit consensus on the final approach.
There was a correction made to the original equation, which was initially misstated. The discussion also highlights the importance of retaining the original problem for educational purposes.
sorry I made a mistake in the original equation and now its fixed.andrewkirk said:The First term in your 2nd line is wrong, and the error propagates through from there.
what do you mean by left hand side ?andrewkirk said:You've expressed the right-hand side in terms of v, so that's progress. Now you need to do similar work on the left hand side.
Using ##v=\frac{y}{x}##, express ##\frac{dv}{dx}## in terms of y, y' and x, then see if you can use that to express y' in terms of v, v' and x. Equate that to the RHS and then with any luck you'll be able to use separation of variables to get an expression to be integrated over x on one side and the same for v on the other.
hey man, thanks for your time and help, I was able to get it resolved and it matched the answer in my book.andrewkirk said:LHS of the diff equation in section 1 of your OP, which is: ##\frac{dy}{dx}##.