Final answer:
To solve the initial-value problem, we must separate variables, integrate both sides, and then apply the initial condition to find the constant of integration. The final explicit solution is given by y = tan(-1/(x²) + π/4) / 2.
Step-by-step explanation:
To find an explicit solution of the given initial-value problem, (1 x⁴) dy / dx = x(1 + 4y²), y(1) = 0, we need to solve this first-order differential equation. To start, we can separate the variables and integrate both sides. The differential equation can be rewritten as dy / (1 + 4y²) = (x/x⁴) dx, which simplifies to dy / (1 + 4y²) = dx / x³. Integrating both sides gives us ∫ dy / (1 + 4y²) = ∫ dx / x³, leading to arctan(2y) / 2 = -1/(2x²) + C, where C is the constant of integration. Using the initial condition y(1) = 0, we can find that C = π/4. The explicit solution to the differential equation is then y = tan(-1/(x²) + π/4) / 2.