## experimentX 3 years ago solve: $x\; {\partial z \over \partial x} + y\; {\partial z \over \partial y} = z$ that passes though $$x^2+y^2+z^2=25$$ and $$x+y=1$$

1. experimentX

the answer according to book is $25(x+y) = x^2+y^2+z^2$ looks like intersection of two surfaces more than solution of DE

2. experimentX

the solution looks like this ...

3. experimentX

what kind of surface is that ... man i thought it would be a circle.

4. mukushla

man how can we solve something like this? from the $$x^2+y^2+z^2=25$$ we have ${\partial z \over \partial x}=-\frac{x}{z}$${\partial z \over \partial y}=-\frac{y}{z}$put back in the original equation$x^2+y^2=-z^2$lol

5. experimentX

i'm supposed to do it by charpit's method ... an charpit's method is supposed to be easy than lagrange's method ... lol

6. hartnn

i exactly did the same thing and got the same result!! x^2+y^2+z^2=0 !!

7. mukushla

but this is wrong because it gives x=y=z=0

8. experimentX

The lagrange method works ... but this is too ugly ... I have to eliminate z from 4th order equation

9. experimentX

this is my last problem from first order DE ... I'm moving on to second order DE after this Q http://www.mathresources.com/products/mathresource/maa/charpits_method.html

10. mukushla

man i cant get that answer with charpit

11. mukushla

@experimentX santosh where are u....lol

12. experimentX

still here man ... fishing answer from another site.

13. experimentX

what did you get for answer? i got z = a x + phi(a) y + c

14. experimentX

had been doing $z = k_1x+k_2y+k_3$

15. experimentX

k2 should be some function of k1 ... i guess there it would reduce my trouble by half

16. mukushla

man let me try again..

17. experimentX

let me try it again too

18. Valpey

The intersection of the sphere and the plane will be a curve. The equation of the curve will solve $x^2+(1-x)^2+z^2=25$

19. experimentX

I need to find the particular integral of the given DE passing through both of these curves.

20. experimentX

no luck ... can't find $$\phi(k_1)$$

21. Valpey

$\frac{\partial z}{\partial{x}}=\frac{2x-2(1-x)}{-2z}=\frac{2x-1}{-z}$Similarly solve for y and $\frac{\partial y}{\partial{z}}$

22. experimentX

and substitute it there?

23. Valpey

Sure

24. experimentX

Looks like I messed up with Lagrange sol ... it didn't seem that difficult $z/x = \phi(y/x)$

25. experimentX

the problem reduced to $z = ax +by \\ x^2+y^2+z^2=25\\ x+y=1$ Need to eliminate 'a' and 'b' from these equations.