Solving differential equation, given initial condition

Question

I'm stuck on this problem:

$$\frac{dy}{dx}=\frac{\ln(x^3(y^2+1))-2\ln(\sqrt{(y^2+1)}(x))}{yx}$$

given y(1)= 2.

I tried separating the variables using log rules to get this: $$ yx\frac{dy}{dx} = \ln \frac{x^3(y^2+1)}{(\sqrt{y^2+1}(x))^2}$$

Not sure what I can do next...

Thanks.

Answer 1

HINT:

$$\implies xy\frac{dy}{dx}=\ln x\iff y\ dy=\frac{\ln x\ dx}x$$

Integrate by setting $\ln x=u$ in the Right Side