Question

Solve the differential equation: \[ x^2y \, dx - (x^3 + y^3) \, dy = 0. \]

Hint:

When faced with a separable differential equation, attempt to isolate terms involving \(x\) and \(y\) on opposite sides and integrate each side. In more complicated cases, substitution might simplify the equation.

Answer 1

We are given the differential equation: \[ x^2y \, dx - (x^3 + y^3) \, dy = 0. \] Rearrange the terms to separate variables: \[ x^2y \, dx = (x^3 + y^3) \, dy. \] Now, divide both sides by \(x^2y(x^3 + y^3)\): \[ \frac{dx}{x^3 + y^3} = \frac{dy}{x^2y}. \] This equation is separable. To proceed with solving, integrate both sides: \[ \int \frac{dx}{x^3 + y^3} = \int \frac{dy}{x^2y}. \] However, this integral may require a more advanced method (substitution or numerical solution) depending on the complexity of the functions involved. We can express the general solution as: \[ F(x, y) = C, \] where \(F(x, y)\) is a potential function derived from the integrals, and \(C\) is the constant of integration.