Answer
$$ w=\tan(k\ln x+\pi/4) .$$
Work Step by Step
By separation of variables, we have
$$\frac{dw}{1+w^2}=\frac{k}{x} dx$$
then by integration, we get
$$ \tan^{-1}w=k\ln x+c \Longrightarrow w=\tan(k\ln x+c) .$$
Now, since $y(1)=1$, then $c=\pi/4$.
So we have $$ w=\tan(k\ln x+\pi/4) .$$
