Question

A differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. $$ \frac{d y}{d x}=\frac{2}{\sqrt{25-x^{2}}}, \quad(5, \pi) $$

Short answer

The particular solution to the differential equation is \(y = 2\sin^{-1}(\frac{x}{5})\). Checking it graphically validates the result.

Step-by-step solution

Sketch Two Approximate Solutions

First, draw a slope field, with the slope of each individual 'arrow' of the slope field given by the differential equation \(\frac{d y}{d x}=\frac{2}{\sqrt{25-x^{2}}}\). Draw two curves on the slope field, one passing through the given point (5, π). These curves are the visual representation of the solutions to the differential equation.

Integrate the Differential Equation

To determine the particular solution to the differential equation, integrate the following equation: \(\int dy = \int \frac{2}{\sqrt{25-x^{2}}} dx\). The antiderivative is \(y = 2\sin^{-1}(\frac{x}{5}) + C\), where C is the constant of integration.

Find the Constant of Integration

Substitute the given point, (5, π), into the solution and solve for C. Thus, \(π = 2\sin^{-1}(1) + C \Rightarrow C = π - 2 \times \frac{ π}{2} = 0\). Therefore, the particular solution is \(y = 2\sin^{-1}(\frac{x}{5})\).

Graph the Solution Using a Graphing Utility

Use a graphing utility to graph the particular solution \(y = 2\sin^{-1}(\frac{x}{5})\). Compare this graph with the sketch of the approximate solutions made in step 1.

Print the Graph

For a more detailed analysis, print an enlarged copy of the graph from the website www.mathgraphs.com and verify if the original sketches resemble the actual solution.