91Ó°ÊÓ

Differential equations are mathematical equations that relate a function with its derivatives. In our problem, we started with the differential equation \(\frac{d x}{d p}=\frac{10}{\sqrt{p-3}}\). This equation connects the rate of change of \(x\) with respect to \(p\). To solve it, we focus on finding the function \(x\) that satisfies this relationship.
Understanding differential equations is crucial because they model how things change and grow. They are used in various fields like physics, engineering, and economics. When dealing with such equations, we often convert them into integrable forms to solve them. In this case, we rearrange it to \(dx = \frac{10}{\sqrt{p-3}} dp\), paving the way for integration.

Integration is a fundamental concept in calculus that essentially reverses differentiation. When you have a derivative (as in our problem's differential equation), integration helps you find the original function, or at least a family of functions. For solving the given equation, \(\int dx = \int \frac{10}{\sqrt{p-3}} dp\), we integrate on both sides.
The integration results in \(x = 20 \sqrt{p-3} + C\), which indicates that \(x\) is related to \(p\) through the square root function, plus a constant \(C\). This has a crucial role in determining how supply behaves with changing \(p\). The constant \(C\) represents any initial conditions not accounted for in the integration alone, emphasizing the need to consider all initial constraints to fully solve the equations.

Initial conditions are necessary to determine the specific solution from a family of solutions given by the general equation\(x = 20 \sqrt{p-3} + C\). These are conditions that allow us to find the particular constant term \(C\) in the integration process. For instance, we used \(x=100\) when \(p=3\) to find \(C\).
This specificity ensures we get the exact solution for the supply function rather than a general one. By substituting these values, we calculated that \(C = 100\). Initial conditions act as the additional information needed to tailor the solution to fit the particular scenario being modeled, thus helping build a complete picture.