Question

Differential:solve for y
yy'=y+sqrt(x^2+y^2)

Answer 1

i wonder, let y = vx
y' = v'x + v

might be useful

Answer 2

So originally i tried to do that and i got to

Answer 3

one second

Answer 4

\[x \frac{ dv }{ dx }=1-v+\sqrt{\frac{ 1 }{ v^2 }+1}\]

Answer 5

from here would i try to do separable?

Answer 6

yy'=y+sqrt(x^2+y^2)

vx(v'x+v)=vx+sqrt(x^2+v^2x^2)

vv'x^2+v^2x = vx+x sqrt(1+v^2)

v'+v/x = 1/x+ sqrt(1+v^2)/vx

v' = 1/x+ sqrt(1+v^2)/vx - v/x

v' = 1/x (1 + sqrt(1+v^2)/v - v)

dv / (1 + sqrt(1+v^2)/v - v) = dx /x

hmmm, can we make that left side any better?

Answer 7

\[\frac{v}{v-v^2+\sqrt{1+v^2}}dv=\frac1xdx\]
im wondering of a conjugate would be useful .... these are just ideas, how useful they may be is anyones guess

Answer 8

alright sounds good..ill play with it for a little while and see what i get

Answer 9

burnooli comes to mind for non linear diffy Qs, but i cant say a recall much about it

Answer 10

i really do need to bruch up on my diffy Qs :)

Answer 11

Ehh bummers i'm not getting the "simple, clean" answer that they have written. I'll just go into office hours, but thank you for your help