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Integral curves are a fundamental concept when dealing with differential equations. They represent the family of curves that satisfy a given differential equation. Each curve illustrates how the function behaves across different conditions in the plane. For instance, if we consider the differential equation given by \(y' = \frac{y}{2x}\), we're essentially looking at how the slope, \(y'\), changes based on not just \(x\) but also \(y\).
When sketching integral curves, we interpret the given differential equation to understand how each curve might look. These curves are generated by the solution of the differential equation, and each specific curve corresponds to a set of initial conditions. In the exercise, sketching involves finding integral curves that meet specified conditions. Specifically, these curves give us visual insights into how solutions to the differential equation morph as initial conditions vary. Integral curves provide an intuitive graphic representation of solutions to differential equations and often help verify analytical solutions.

A separable differential equation is a type of differential equation in which variables can be separated and rewritten such that all terms involving one variable are on one side of the equation and all terms involving another variable are on the opposite side. The ability to separate variables is crucial because it allows us to integrate each side independently.
For the differential equation \(y' = \frac{y}{2x}\), separation of variables involves rewriting it as \(\frac{dy}{dx} = \frac{y}{2x}\). We can rearrange this to \(\frac{dy}{y} = \frac{1}{2x} dx\). By separating the equation in this way, each side is now ready for integration, leading to individual integrals that can be solved step-by-step.
This technique is particularly useful because it simplifies complex problems into more manageable ones. Once separated, each integral can be calculated, eventually leading to a general solution for the given differential equation. Separable differential equations are commonly encountered in many fields of science and engineering, making this method a valuable tool for solving a variety of practical problems.

When integrating to solve a differential equation, a constant of integration often appears as part of the solution. This constant arises because indefinite integration yields a family of functions, each differing by a constant. For instance, when integrating \(\int \frac{dy}{y} = \ln |y|\), we add an arbitrary constant \(C\) to account for these infinite possibilities.
In the steps given for solving the differential equation, after integration, the solution was \(\ln |y| = \frac{1}{2} \ln |x| + C\). This \(C\) is the constant of integration that ensures all potential solutions are accounted for. The constant is crucial because it allows us to pinpoint specific solutions based on initial conditions—like finding which particular curve passes through a specified point \((2, 1)\).
Determining the value of the constant of integration involves substituting known values into the solved equation to find a particular solution. This value, once calculated, refines the general solution into one unique to the initial conditions provided. In the exercise, solving \(1 = C_1 \cdot 2^{1/2}\) led to finding \(C_1 = \frac{1}{\sqrt{2}}\), thus enabling the precise integral curve equation, \(y = \frac{1}{\sqrt{2}} \cdot x^{1/2}\). Such specificity is necessary for accurately describing the behavior of a system modeled by the differential equation.