Question

solve y^2 dy = x(x dy - y dx)

Answer 1

(DE)

Answer 2

@DLS

Answer 3

@paki

Answer 4

homogeneous DE, so just plug in
\(\Large y = vx \\ \Large dy = vdx +x dv \)

simplify and it will get converted into easily separable DE, try it :)

Answer 5

when should we use y=vx and x=vy?

Answer 6

non-homogeneous DE
both works.

Answer 7

we can use them for non homo and for homo. DE?

Answer 8

no, i meant you can use both

y=vx and x=v

but for only homogeneous eq.
For non-homogeneous eq, different metod are there

Answer 9

i got
\[\ln \frac{ x ^{2} }{ y } = C + \frac{ x ^{2} }{ 2y ^{2} }\]

???

Answer 10

i think ln x is getting cancelled....
can you check your work again ??

or show your attempt, i will help you spot the error :)

Answer 11

y^2 dy = x(x dy - y dx)
dy(y^2-x^2)=-xy dx
\[\LARGE \frac{dy}{dx}=\frac{xy}{x^2-y^2}\]
put y=vx
\[\LARGE v+ x \frac{dv}{dx}=\frac{v}{1-v^2}\]
\[\LARGE x \frac{dv}{dx}=\frac{v}{1-v^2} - \frac{v(1-v^2)}{1-v^2}\]
\[\LARGE x \frac{dv}{dx}=\frac{v^2}{1-v^2}\]
\[\LARGE \frac{1-v^2 }{v^2} dv = \frac{dx}{x}\]
Now integrate both sides and compare the starting steps I did with urs.

Answer 12

why u do dis dls ;-;

Answer 13

i didn't write the whole solution hartnn,just the starting steps

Answer 14

uhmm mine is