91Ó°ÊÓ

Integrating differential equations is a foundational technique in calculus used to find a function when its derivative is known. It involves reversing the process of differentiation, which can be interpreted as finding the area under the curve of a given function's rate of change. In the context of our exercise, we are given the slope of the tangent line at any point on the function's graph, which is the derivative, \(\frac{dy}{dx} = \frac{3x^2}{2y}\).

To integrate this differential equation, we first rewrite it in a way that separates the variables (more on this in the next section). After separation, we're left with integrals on both sides of the equation, one with respect to \(x\) and the other with respect to \(y\).

Upon integration, we obtain an expression involving the function \(y\) and the integration constant \(C_1\). Integration constants are crucial as they represent the 'family' of possible functions that satisfy the original differential equation. The next steps involve finding the specific constant for our function by using the given initial condition.

The slope of the tangent line to a curve at any given point represents the instantaneous rate of change of the function at that point. It is effectively the derivative of the function at that specific point on its graph. Differential equations often involve these slopes, as seen in the exercise, where the differential equation relates the slope of the tangent line at any point \((x, y)\) on the graph to the variables themselves.

Mathematically, if \(f(x)\) is a function, then the slope of the tangent line at \(x\) is given by the derivative \(f'(x)\). In this problem, the slope given is \(\frac{3x^2}{2y}\), so finding \(f(x)\) requires solving this specific differential equation where the slope, inherently connected to the concept of the derivative, varies with both \(x\) and \(y\).

The separation of variables is a technique for solving differential equations where we manipulate the equation to isolate one variable and its differential on one side, and the other variable and its differential on the other side. Here's how this process looked in our exercise:
\[2y dy = 3x^2 dx\]

By separating \(x\) terms on one side and \(y\) terms on the other, we set up the equation to integrate both sides independently. This creates two separate integrals that can be solved using standard calculus techniques. The goal of separation of variables is to simplify a complex differential equation into a form where we can easily integrate to find the relationship between \(x\) and \(y\).

When we integrate a function, we often add an arbitrary constant, known as the integration constant. This is because indefinite integration is the reverse of differentiation, and when differentiating a constant, the result is zero—hence, the constant 'disappears' during differentiation. For our exercise, after integrating the separated variables, we introduced an integration constant \(C_1\).

The integration constant plays a vital part in the solution of differential equations. It represents an infinite number of possible functions that satisfy the differential equation. However, we can find the exact value of this constant using initial or boundary conditions. In our exercise, we used the point \((1,3)\) to solve for \(C_1\), thus pinpointing the specific function that not only satisfies the differential equation but also passes through the given point on the graph.