Expert: Paul Klarreich - 12/13/2010

Question
x^2*y'=x^2+y^2

Answer
Questioner: Kaira
Country: Ukraine
Category: Calculus
Private: Yes   <<< changed.
Subject: differential equation
Question: x^2*y'=x^2+y^2
...................................
x^2 y'=x^2+y^2

Divide:

 y'=  1+y^2/x^2

Separate it into two parts:

y' = 1  -->  y = x + C1

The other part is:

 y'=  y^2/x^2

 dy/dx = y^2/x^2

Do a separation of variables:

dy    dx
--- = ---
y^2   x^2

-1   - 1
--- = --- + C2
y     x

1     1
--- = --- - C2
y     x

1     1 - C2 x
--- = ---------
y       x

       x
y = --------
   1 - C2 x

Now add in the other part and you have:

       x
y = --------  + x + C1
   1 - C2 x