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The slope of a curve at a given point represents how steep the curve is at that specific point. It tells you the rate of change of the dependent variable, usually denoted as \(y\), as the independent variable, typically \(x\), changes. In mathematical terms, this is expressed as the derivative \(\frac{dy}{dx}\). For the problem at hand, the slope at each point on the curve is given as \(3\sqrt{x}\). This means wherever you pick a point \(x\) along the curve, the slope there would reflect the steepness by the factor of \(3\) multiplied by \(\sqrt{x}\). When you see an expression like this, it hints at how the curve behaves. The slope being dependent on \(\sqrt{x}\) indicates that as \(x\) becomes larger, the change in \(y\) gets amplified since \(\sqrt{x}\) increases.
This concept is crucial for understanding how to set up and solve differential equations involving real-world problems.

The initial condition in a differential equation gives us the precise 'starting point' of the curve on a graph. It's a known value of the function at a specific point that helps us find unknown quantities in the equation. Without an initial condition, a differential equation can have many solutions due to the constant of integration.
In the exercise, we have a curve passing through the point \((9,4)\). This tells us that when \(x = 9\), \(y\) must be \(4\). With this piece of information, we can substitute into our integrated equation to solve for the constant \(C\).
Think of it like using a GPS to find your location. You need a starting point to map the rest of the route effectively. In mathematics, initial conditions serve the same role by helping us locate the exact path our solution curve will take.

Integration is the process of finding the function that describes a curve, given its slope. In the context of differential equations, it typically follows setting up the differential equation. In our example, the slope was \(3\sqrt{x}\). Integration tells us: Given this rate of change, what was the original function?
To integrate \(\frac{dy}{dx} = 3\sqrt{x}\), we find \(y\) by integrating both sides with respect to \(x\). This means solving \(\int 3 \sqrt{x} \, dx\). We used the power rule for integration, letting us change \(x^{1/2}\) to \(x^{3/2} \frac{2}{3}\). This results in \(2x^{3/2}\) after applying the coefficient \(3\). But integration involves an unknown constant, which we denote with \(C\).

• The integral of a constant slope on the left (\(dy\)) is simply \(y\).
• The right side becomes \(2x^{3/2} + C\).

Through integration, we transmute the differential equation back into its parent function.

When we perform integration, especially indefinite integration like in our exercise, it introduces a constant of integration, represented as \(C\). This occurs because the derivative of any constant is zero, so integrating doesn't initially inform us of any constant terms that may have been present in the original function.
The constant \(C\) is crucial because it fine-tunes our integrated function to satisfy any particular conditions or specific data points. In the example given, the requirement that the curve pass through \((9,4)\) allows us to calculate \(C\). We substitute \(x=9\) and \(y=4\) into the integrated equation \(y = 2x^{3/2} + C\) to solve for \(C\).
By doing this calculation, it became clear \(C = -50\). Having this constant sets our solution apart from infinitely many others, and allows our function to meet the specific requirements given by the initial condition, ensuring our solution accurately describes the real scenario.