Final answer:
The solution involves separating variables and integrating both sides, then applying the initial condition y(0) = 1/2 to find the particular solution to the initial-value problem.
Step-by-step explanation:
To find an explicit solution to the given initial-value problem 1 - y² dx - 1 - x² dy = 0, with the initial condition y(0) = 1/2, the problem can be approached by separating variables. Start by adding 1 - x² dy to both sides to isolate terms involving dx and dy. Separating variables gives us (1 - y²)dx = (1 - x²)dy. Dividing both sides by (1 - y²)(1 - x²) and then integrating both sides yields the solution. Use partial fractions if necessary to simplify the integration process.
To satisfy the initial condition y(0) = 1/2, substitute x = 0 and y = 1/2 into the general solution and solve for any constants that may appear during integration. This process yields the particular solution to the initial-value problem.
Please note, additional context or specific methods of integration may be needed to complete the solving process, depending on the complexity of the integral that arises after separation.