Separable Differential Equation with Initial Condition

Solve the differential equation $\frac{dy}{dx} = \frac{x^2}{y^2}$ using separation of variables to find the general solution and the particular solution given the initial condition $y(1) = 2$.

In this problem, we are tasked with solving a first-order differential equation using the method of separation of variables. This technique is often applied when a differential equation can be rewritten such that all terms involving the dependent variable and its differentials are on one side of the equation, and all terms involving the independent variable are on the other side. This separation allows us to integrate both sides individually, leading to a solution. The goal here is not only to find a general solution but also to determine a particular solution given a specific initial condition.

The general solution provides a family of functions that satisfy the original differential equation, whereas the particular solution is one specific member of this family that satisfies an additional condition known as an initial condition. An essential concept in dealing with differential equations is understanding how these initial conditions uniquely determine a solution, which is frequently encountered across various physical applications, such as in dynamics, growth processes, and electrical circuits.

In addition to practicing the technique of separation of variables, this problem reinforces the importance of constant of integration that arises when solving indefinite integrals during the process. It highlights how these constants adjust based on computations to satisfy initial conditions. Such differential equation problems serve as foundational examples in courses covering calculus and applied mathematics, thereby forming a critical strategy in understanding more complex systems described by differential equations.

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