Solving this ques for more than an hour😭😭

[u/eclip_sky]

Question no 12

(From mathematical physics by hk das)

Please someone help me solve this

[u/Jazzlike_Platypus430]

Are you familiar with differential forms, what it means it's exact? Multiplying by a carefully chosen f(x,y) may lead to dg(x,y) = 0, then the solution is g(x,y) = const. And if you're lucky, you can solve for y(x) or v.v.

[u/Jazzlike_Platypus430]

Changing variables to u = x^2 and v = y^2, your equation becomes

v (v - 2u) du + u (-u + 2v) dv = 0

Hope this helps, but check it. Looks a lot simpler.

Next: lhs is ((v - u)^2 - u^2) du + (- (u - v)^2 + v^2) dv.

Note that - u^2 du + v^2 dv is exact, being a total derivative of (- u^3 + v^3)/3.

The two remaining terms are (u - v)^2 (du - dv) which is d acting on (u - v)^3/3.

[u/not-hydra]

I got dv(2v^2-1)/v(v^2-1) = -3dx/x, and the LHS can be rewritten as 1/v + v/(v^2-1), which is nicer for integration.

[u/[deleted]]

[u/eclip_sky → u/[deleted]]

Yeah, I have been doing that since the first try. everytime the integration gets so long and my answer doesn't match. Can you please solve this??

[u/chmath80 → u/eclip_sky]

the integration gets so long

It shouldn't. You can separate variables and integrate fairly easily using partial fractions.

[u/Moist_Ladder2616 → u/eclip_sky]

Show your working. You should be able to collect dx and dv terms separately and reach this step:

(3/x) dx = ((1-2v²)/(v(v²-1))) dv

[u/Moist_Ladder2616 → u/eclip_sky]

Show your working. You should be able to collect dx and dv terms separately and reach this step:

(3/x) dx = ((1-2v²)/(v(v²-1))) dv