Hi I am trying to solve the following differential equation but not having much luck!
dy/dx = 1+y^2 / 1 + x^2
Its a multiple choice question and the answer is one of the following:
(A) Cx / (1 - Cx)
(B) Cx / (1 + Cx)
(c) (C - x) / (1 - Cx)
(D) (1 - Cx) / (x + C)
(E) (x + C) / (1 - Cx)
Here is what I have so far:
Rearange to the form:
1/(1+y^2) dy/dx = 1/ (1+x^2)
Integrate both sides with respect to x, giving:
tan^-1(y) = tan^-1(x) + c
which is the same as:
y = tan(tan^-1(x) + c)
y = x + tan(c)
we can let C = tan(c) as it is constant, giving:
y = x + C
but this looks nothing like the answer.
Any help much appreciated, have spent ages trying to solve this!
Many thanks.
y = \tan(tan^-1(x) + c)
Using:
we get:
tan(tan^-1(x)) + tan(c) / 1 - tan(tan^-1(x)) * tan(c)
which gives us:
x + tan(c) / 1 + x.tan(c)
if we now let C = tan(c)
we have:
x + c / 1 - Cx
I obviously need to know my Trig Cheat sheet better!
Cheers,
Alex