General solution of a differential equation in implicit form?

Question

if f(x) = allsqrt 10+x^2

is

x / allsqrt 10+x^2

then find the general solution of the differential equation:

dy/dx = (allsqrt 10+y^2) / y (x>0, y>0)

giving the solution in implicit form.

Answer 1

dy/dx =sqrt(10+y^2)/y

ydy/sqrt(10+y^2)=dx

d(10+y^2)=2ydy

∫ydy/sqrt(10+y^2)=∫dx

1/2*∫d(10+y^2)/sqrt(10+y^2)=dx

1/2*∫dv/√v=v+c

sqrt(10+y^2)=x+c

y²=(x+c)²-10