Final answer:
To solve the initial value problem, the differential equation must be separated into variables and integrated, after which the initial condition is applied to find the particular solution.
Step-by-step explanation:
The question requires solving the differential equation with the given initial value y(1)=1. To solve the initial value problem x²dy/dx = (4x²-x-2)/(x+1)(y+1), we first notice that it is a separable equation. Separating variables and integrating both sides, we impose the initial condition to find the particular solution that satisfies y(1)=1. This process involves integration and the application of logarithmic and exponential rules to isolate y as a function of x.