Solve the initial-value problem for the homogeneous differential equation: $x y^2 \frac{dy}{dx}=y^3-x^3$, with $y\left(1\right)=3$. What is the explicit solution?

• A. $y^3-x^3=0$
• B. $y^3+x^3=28$
• C. $y^3-x^3=28$
• D. $y^3+x^3=0$

Answer: (B)

Verified step by step guidance

1. Step 1: Start by rewriting the given differential equation x y^2 (dy/dx) = y^3 - x^3. Divide through by y^2 (assuming y ≠ 0) to simplify the equation to dy/dx = (y^3 - x^3) / (x y^2).
2. Step 2: Separate the variables to prepare for integration. Rearrange the equation so that all terms involving y are on one side and all terms involving x are on the other side. This gives (y^2 / (y^3 - x^3)) dy = (1/x) dx.
3. Step 3: Integrate both sides of the equation. For the left-hand side, integrate with respect to y, and for the right-hand side, integrate with respect to x. Use substitution techniques if necessary to handle the integration.
4. Step 4: After integration, solve for the constant of integration using the initial condition y(1) = 3. Substitute x = 1 and y = 3 into the resulting equation to find the constant.
5. Step 5: Use the constant of integration to write the explicit solution to the differential equation. Simplify the equation to express y explicitly in terms of x, ensuring the solution satisfies the initial condition.