Question

Find the general solution to the following separable DE:
dy/y^2 = 2xdx/sqrt(1+2x^2)

Answer 1

\[\frac{ dy }{ y^2 } = \frac{ 2xdx }{ \sqrt{1+2x^2} } \]

Answer 2

you have to integrate.
For the right part, let \(x^2=t \) and \(2xdx=dt\)
then it will become:
\(\int\huge\frac{dt}{\sqrt{1+2t}}\). Now should be straight forward

Answer 3

after that i integrate the rhs and lhs ?

Answer 4

integrate both sides and arrange it into the general form where y = f(x) + C
if you were not given initial conditions, then that is your solutiopn.

Answer 5

can i have detailed steps?, you guys just give me an overview

Answer 6

It just depends how your professor would like to see the answer.
All you do is integrate both sides. You will get f(y) + k = f(x) + c, but the constant on the other side doesnt matter because when you move it to the other side, it will simply get absorbed by the other constant, so you can simply just write f(y) = f(x) + C
i wrote f(y) and f(x) because you will get some function in terms of y and some function in terms of x.

Does that make sense?" I'm assuming that if you're working with first order seperable differential equations, you should be very comfortable with integrals. Follow what @myko has told you.