## anonymous 4 years ago 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

1. JamesJ

It's not. And the answer should remind you of this trig identity that will be very useful: $\tan(a+b) = \frac{ \tan a + \tan b}{1 - \tan a . \tan b}$

2. cristiann

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 ...