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Separable differential equations are a specific type of first-order differential equation. They allow us to separate the variables into two distinct groups, each with its integrable expression. This separation simplifies the process of finding a general solution to the differential equation.
In general, a differential equation is separable if it can be rearranged into the form:

• \( N(y)\, dy = M(x)\, dx \)

Where \( N(y) \) is a function solely of \( y \), and \( M(x) \) is a function solely of \( x \). In the given problem, \( \frac{dy}{dx} = x y^2 \), the rearrangement yields \( \frac{1}{y^2}dy = x dx \). This separation is crucial because it allows us to integrate both sides independently. It simplifies the process and makes solving such equations straightforward.
By integrating each side with respect to its respective variable, we effectively isolate \( y \) in terms of \( x \) or vice versa, leading us closer to the solution.

A first-order differential equation involves the first derivative of a function, but no higher-order derivatives. The general form is \( y' = f(x, y) \), indicating how the derivative \( y' \) (or \( \frac{dy}{dx} \)) relates to \( x \) and \( y \). Knowing that a differential equation is of first order is important as it often dictates the methods used to solve it.
In the exercise \( y' - xy^2 = 0 \), we identify it as a first-order equation because it only contains the first derivative \( y' \). The absence of higher derivatives makes it a good candidate for specific methods like separation of variables. This understanding allows us to apply appropriate techniques, ensuring we solve the equation efficiently and accurately.

Integration is a fundamental tool required to solve separable and first-order differential equations. When you separate variables within a differential equation, you are often left with two sides that each need to be integrated. Two standard techniques of integration applied here are:

• Basic Antiderivatives: Recognizing the integral forms, such as \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), necessary for the integration of power functions.
• Initial Conditions: Utilization of specific conditions given in a problem (such as \( y(1) = 2 \)) to find the constant of integration, ensuring a unique solution rather than a general one.

For our specific problem, upon integration of \( \frac{1}{y^2} dy \), we get \( -\frac{1}{y} \), and \( x dx \) results in \( \frac{x^2}{2} + C \). Integrating these expressions allows us to solve for \( y \) explicitly, using the initial condition to determine the exact value of \( C \). Such techniques are quintessential as they provide the link between forming an integrated equation and deriving a specific, usable solution to a differential equation.