Solve the differential equation dy∕dx = (y²+4)/(x²+16), y(4)=1.

Hello!
This differential equation is a separable one, it is possible to separate y from x. For this, simply divide both sides by (y^2+4):
(y')/(y^2+4) = 1/(x^2+16).
y is at the left side only, x is at the right side only. Moreover, both sides are integrable in elementary functions:
1/2 arctan(y/2) = 1/4 arctan(x/4)+C,
or  arctan(y/2) = 1/2 arctan(x/4)+C.  (1)
 
Now use the given boundary condition, y(4)=1, to find C:
arctan(1/2) = 1/2 arctan(1) + C, or
C =arctan(1/2) - 1/2 arctan(1) = arctan(1/2) - pi/8.
 
If we take tan of the both sides of (1), we obtain
y(x)=2tan(1/2 arctan(x/4)+arctan(1/2)-pi/8).
This is the only solution.