Olá, Deivid.
[tex]\text{Fazendo }\frac{y}{x}=u,\text{ temos:}\\\\ y=ux \Rightarrow \frac{dy}{dx} = \frac{du}{dx}\cdot x + u\\\\ \text{Substituindo na equa\c{c}\~ao, temos:}\\\\ x \cdot \frac{du}{dx} + u = u + \tan u \Rightarrow x\,du = \tan u\,dx \Rightarrow \frac1{\tan u}\,du =\frac1{x}\,dx \Rightarrow\\\\ \int{\cot u\,du = \int{\frac1{x}}\,dx \Rightarrow \ln |\sin u| = \ln |x| \Rightarrow \sin u = x \Rightarrow[/tex]
[tex]u = \arcsin x \Rightarrow \frac{y}{x} = \arcsin x \Rightarrow \boxed{y(x) = x \cdot \arcsin x}[/tex]
