Question

y'=(1+y^2)/(1+x^2)
y(2)=3
book answer
y=(x+c)/(1-cx)

how is this possible with out trig substitution

Answer 1

It's not. And the answer should remind you of this trig identity that will be very useful:

\[ \tan(a+b) = \frac{ \tan a + \tan b}{1 - \tan a . \tan b} \]

Answer 2

y′=((1+y²)/(1+x²)), y(0)=3
Equation with separable variables:

((y′)/(1+y²))=(1/(1+x²))⇒arctan y=arctan x+C
arctan y(0)=arctan0+C⇒arctan3=C
arctan y=arctan x+arctan3
[this is the solution in implicit form... up to here the differential equation problem should be considered solved]
To solve the implicit function problem, use the formula JamesJ gave you:
⇒y(x)=tan(arctan x+arctan3)=((x+3)/(1-3x))
Please note this is just the first-level solution ... strictly speaking, you still have to find the (maximal) domain of the solution, maybe discuss some other issues ...