Final answer:
The solution involves rewriting the differential equation in a separable form, integrating both sides, and applying the initial condition y(1) = 4 to find the constant of integration, which leads to the particular solution.
Step-by-step explanation:
To solve the given initial-value problem xy² dy/dx = y³ - x³, first rewrite it in a separable form by dividing both sides by xy², yielding dy/dx = (y/x) - (x²/y²). Next, separate variables and integrate both sides. Applying the initial condition y(1) = 4 to find the constant of integration will give us the particular solution to the differential equation.
The process involves integration and substituting initial values into the resulting general solution to find the specific solution that satisfies the initial condition. Without the actual integration and algebraic manipulation shown, the step-by-step solution would be incomplete.
It is important to note that any solution to a homogeneous differential equation can be verified by substitution into the original equation. If you think of y as a function of x, i.e., y = y(x), you can differentiate this function with respect to x and replace dy/dx in the original equation to confirm correctness. For this specific task, the student would need to follow through the steps of separation, integration, and application of the initial condition to find the particular solution that also satisfies the given condition y(1) = 4.