Final answer:
To solve the given initial value problem, use the method of separation of variables and integrate both sides with respect to y and x. Set the two sides equal to each other and substitute the initial condition to find the constant of integration. The solution to the initial value problem is x^4 = 1/4.
Step-by-step explanation:
To solve the given initial value problem, we can use the method of separation of variables. First, rearrange the equation to have the dy/dx term on one side and all other terms on the other side. This gives us: y^2 dy = (y^3-x^3) dx. Next, integrate both sides with respect to y and x. On the left side, we integrate y^2 dy to get (1/3) y^3. On the right side, we integrate (y^3-x^3) dx to get (1/3) y^3 - (1/4) x^4 + C. By setting the two sides equal to each other, we have (1/3)y^3 = (1/3)y^3 - (1/4)x^4 + C. Simplifying this equation, we get x^4 = C. Now, substitute the initial condition y(1) = 4 into the equation to find C. Plugging in x = 1 and y = 4, we get (1/3)(4^3) = (1/3)(4^3) - (1/4)(1^4) + C. Simplifying further, we get 64/3 = 64/3 - 1/4 + C. Solving for C, we find C = 1/4. Therefore, the solution to the initial value problem is x^4 = 1/4.