91Ó°ÊÓ

The method of Separation of Variables is a fundamental technique used to solve first-order linear differential equations, where the equation can be manipulated so that each variable appears on opposite sides of the equation.

In our exercise, the differential equation given was \(y' = \sqrt{x} y\). To implement separation of variables, we want to isolate \(y\) and its derivative \(y'\) on one side, and \(x\) with its derivatives on the other. This step involves algebraic manipulation, resulting in \(\frac{y'}{y} = \sqrt{x}\). Now, all instances of \(y\) are on the left and all instances of \(x\) are on the right, ready for integration.

The process of Integration of Functions allows us to find the original functions from their derivatives, which is the reverse operation of differentiation.

In the given problem, after separating variables, we are led to integrate both sides: \(\int \frac{y'}{y} dx = \int \sqrt{x} dx\). Integrating \(\frac{y'}{y}\) with respect to \(x\) yields \(\ln|y|\), the natural logarithm of the absolute value of \(y\), since the derivative of \(\ln|y|\) with respect to \(y\) is \(\frac{1}{y}\). On the right side, integrating \(\sqrt{x}\) with respect to \(x\) gives us \(\frac{2}{3}x^{3/2}\), as the integral of \(x^{n}\) is generally \(\frac{x^{n+1}}{n+1}\).

Integration is a powerful tool in solving differential equations as it helps to reconstruct the unknown function from its rate of change.

The concept of Exponential Functions plays a crucial role in solving differential equations, particularly after taking the integral of each side.

Once we obtain the form \(\ln|y| = \frac{2}{3}x^{3/2} + C\), where \(C\) is the constant of integration, we use the property of logarithms to solve for \(y\). Recognizing that \(e^{\ln|y|} = |y|\), we can convert the logarithmic form into its exponential counterpart, thereby getting rid of the logarithm. This leads to \(y = e^{\frac{2}{3}x^{3/2} + C}\) or \(y = e^{C}e^{\frac{2}{3}x^{3/2}}\), showcasing the exponential function's flexibility in simplifying expressions and solving equations.

When integrating, the Constant of Integration represents an arbitrary constant added to the function to account for the fact that indefinite integration is not unique.

In the equation found through integrating \(\ln|y| = \frac{2}{3}x^{3/2} + C\), \(C\) symbolizes all the possible constants that could be added to the integral to make it a family of functions, rather than a single solution. It is important in the context of initial value problems where a specific solution is required, and the value of \(C\) can be determined when a condition (\(y(x_0) = y_0\)) is given. Without an initial condition, we express the general solution to our differential equation involving \(C\), illustrating that there are infinitely many solutions, each differing by a constant.