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# first order differentials watch

The equation to solve is:

xdy/dx = y + (x^2 - y^2)^(1/2)

any ideas?
- it isn't in the form dy/dx + P(x)y=q(x) the main problem being the q(x) function.
- i have tried integrating factor approach to help separate it...but this is useless given q(x) has y terms in it.
- any suggested substitutions to use?
thanks!
2. What level of maths are you at (A-level/1st year university/2nd year/etc)? I feel like there should be some sort of substitution given here. Are there any previous parts to the question?
3. hi well i'm doing sciences at uni (1st year) and this is the maths course for my course. i thought i could use sec(u) and tan(u) maybe?? that could make the
(x^2 - y^2)^(1/2) function disappear.
4. yes there are previous parts
the full part of the question is:

Solve the following first order differential equations:
i) x dy/dx = 3(1-y) given y=0 and x=1

ii) (1 + x^2)^(1/2)dy/dx=xexp(-y) given y=0 at x=0

iii) xdy/dx = y + (x^2 - y^2)^(1/2)

no boundary/initial conditions are given for iii).
5. it's fine i've got it now - thanks.
6. Divide through by x first to get

dy/dx = y/x + ((x^2 - y^2)^0.5)/x

Then note that

Now sub in u = y/x.

That may or may not help you. I haven't tried it, but I can't see anything else to do.
7. Try .
8. (Original post by ibush1)
it's fine i've got it now - thanks.
What did you use to solve it?
9. (Original post by ibush1)

The equation to solve is:

xdy/dx = y + (x^2 - y^2)^(1/2)

any ideas?
- it isn't in the form dy/dx + P(x)y=q(x) the main problem being the q(x) function.
- i have tried integrating factor approach to help separate it...but this is useless given q(x) has y terms in it.
- any suggested substitutions to use?
thanks!
I used y=xcos u and it seems to reduce to a separable first order.
10. (Original post by Farhan.Hanif93)
Try .
Yeah, this is equivalent to y/x = u. The benefit of not putting sin(theta) in from the start is that the u substitution is a bit easier and the resulting integral can be recognized.
11. (Original post by Swayum)
Yeah, this is equivalent to y/x = u. The benefit of not putting sin(theta) in from the start is that the u substitution is a bit easier and the resulting integral can be recognized.
Fair enough.

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