Solve a Separable Differential Equation

Solve the differential equation $y' = \frac{x^2}{y}$ by showing that it is separable and using integration to find the solution.

This problem introduces a first order differential equation which is separable, an important concept when solving ordinary differential equations. Separable differential equations are of the form where variables can be separated on opposite sides of the equation. This means that one side contains only one variable and its differential, and the other side contains only the other variable and its differential. Solving these problems typically involves algebraic manipulation to reformat the equation, followed by integration which requires basic integration techniques. In this problem, you'll identify how to rearrange the given equation to match the separable format, allowing you to integrate both sides with respect to their corresponding variables.

Integrating the separated equation often involves applying integration rules such as the power rule, logarithmic integration, and potentially substitution if the integral is complex. Understanding how and when to apply these rules is critical as it enables you to find the general solution of the differential equation. Upon integrating, solutions might involve solving for constants of integration, often using initial conditions if they are given. Once you have the integrated form, you may also need to simplify your expression to get the final explicit solution for the differential equation. Conceptually, mastering separable differential equations enhances your ability to analyze and solve more complex differential problems by grounding you in foundational solution methods.