Help required plz...?

Question

Find an implicit and an explicit solution to the initial value problem
dy/dx = (y^(2) - 1) / (x^(2) - 1), y(2) = 2.

Please show all workings - thanks.

Answer 1

dy/dx = (y^(2) - 1) / (x^(2) - 1), y(2) = 2.

dy/(y^2-1)=dx/(x^2-1)
write in simple fractions

{(1/2)/(y-1)}dy
+[(-1/2)/(y+1)]dy=

{(1/2)/(x-1)}dx
+[(-1/2)/(x+1)]dx

integrate both sides
(1/2)ln(y-1)+(-1/2)ln(y+1)
=(1/2)ln(x-1)+(-1/2)ln(x+1)

Hence
(y-1)/(y+1)
=C.(x-1)/(x+1)

for x=2
y=2
C=?

C=1

Then

y=x+1