Question

find the general solution by separating the variables:

x(1+y^2)dx +y(1+x^2)dy=0

Answer 1

x(1 + y^2) = -y(1 + x^2)dy

Answer 2

xdx / (1 + x^2) = -ydy/(1 + y^2)

Answer 3

For the left hand side to take the integral let u = 1 + x^2 then du = 2xdx so du/2 = xdx. Also for the right hand side let u = 1+ y^2 then du = 2ydy which also means that -du/2 = -ydy

Answer 4

So know you have

Answer 5

\[1/2\int\limits_{}^{} du/u = -1/2\int\limits_{}^{} du/u\]

Answer 6

1/2ln|u| = -1/2lln|u|
ln|1+ x^2| = -ln|1 + y^2] + C

Answer 7

Then you take e to the power of both sides:
getting rid of the natural logs:
1+ x^2 = e^c(1 + y^2)...keep in mind that e^c is another constant lets say K.

Answer 8

1/k(1 + x^2) = 1 + y^2
y^2 = 1 - 1/k(1 + x^2)
y = sqrt(1 - 1/k(1 +x^2))

Answer 9

It might also be clearer and more understandable if for the integration you set different variables for your denominators instead of using u for both the left and right hand side.

Answer 10

do you follow?

Answer 11

yea, the problem I had was that I messed up after the separation and made a large mess out of the integration. Everything after the integration is the easier part for me.

Answer 12

Or I should say finding the proper integration method was my problem.

Answer 13

okay yeah with separation of variables dividing and setting up both sides to integrate is key. It's good that you understand though but I assure you that as you do more practice with separation of variables you will master various integration techniques as well.

Answer 14

after I write this one down i'll be trying out a few more and hopefully that is the case since practice should make perfect :)

Answer 15

exactly! that's how I learnt it! but if you have any questions don't hesitate to post them.

Answer 16

okay, thanks alot for helping me out

Answer 17

no problem