Problem 31 - Essential Calculus

Question

Use a computer algebra system to graph the slope field for the differential equation and graph the solution through the given initial condition. $$ \begin{array}{l} \frac{d y}{d x}=\frac{6}{4-x^{2}} \\ y(0)=3 \end{array} $$

Step-by-Step Solution

Answer

The slope field will show vertical asymptotes at \(x = 2\) and \(x = -2\). The solution curve will pass through the point \(0, 3\) and follow the pattern of the slope field.

Step 1: Analyze the Differential Equation

The given differential equation is \(\frac{dy}{dx}=\frac{6}{4-x^2}\). This equation can be rewritten as \(\frac{dy}{dx}= \frac{6}{(2-x)(2+x)}\), clearly expressing the points at which the slope field is undefined, namely \(x = 2\) and \(x = -2\). These are the vertical asymptotes.

Step 2: Graph the Slope Field

Utilize the algebra system's slope field or direction field function, typically titled 'plotdf' or similar, to graph the given differential equation, \(\frac{dy}{dx}=\frac{6}{4-x^2}\). This generates a grid of tiny line segments, each representing the slope of the solution at that point.

Step 3: Evaluate and Apply the Initial Condition

The initial condition given is \(y(0) = 3\). Substitute \(x = 0\) into the differential equation to find the corresponding \(y\) value and plot that point on the graph. This point, \(0, 3\), lies on the solution curve.

Step 4: Graph the Solution Curve

Using the initial condition, plot the solution curve on the slope field. The curve starts at the point \(0, 3\) and should follow the pattern of the slope field. Make sure it comes reasonably close to each of the points already plotted and continues this pattern beyond these points. The solution curve should pass through the provided initial condition.

Key Concepts

Slope Field

A slope field, also known as a direction field, is a visual representation of a differential equation's solutions in a particular plane. It consists of small line segments or arrows at various points in the plane, each illustrating the slope of the solution at that point.

For the equation \(\frac{dy}{dx} = \frac{6}{4-x^2}\), the slope field shows the rate of change of \(y\) with respect to \(x\) for all values except at \(x = 2\) and \(x = -2\), where the equation is undefined. These are asymptotes, crucial in understanding the behavior of the solution.

The segments' direction in a slope field hints at the nature and path of potential solution curves. By observing these lines, one can predict how the solution curve should behave, providing a roadmap for solving the differential equation visually.

Initial Condition

An initial condition is a specific value given to the function or its derivative at a particular point. It helps in pinpointing a unique solution from a family of curves.

In our scenario, the initial condition is \(y(0) = 3\), which means that when \(x = 0\), \(y\) must equal 3. This specific condition serves as a starting point on the slope field.

By plotting this point on a graph, the initial condition ensures that only one trajectory or curve will match, addressing the precise solution of the differential equation rather than a general solution.

Visualizing the initial condition as a plumber working with a tool: the plumber (the equation) knows the general shape but requires the tool (the initial condition) to 'fix' it into place.

Computer Algebra System

A computer algebra system, or CAS, refers to powerful software that assists in solving, analyzing, and visualizing mathematical problems. They are equipped to perform symbolic mathematics and provide intricate solutions that might be cumbersome by hand.

For plotting our differential equation's slope field, a CAS can automate the creation of these visual representations without manual graphing.

• These systems can handle the plotting of derivative functions and create slope fields efficiently.
• Users can use functions like 'plotdf' to observe how solutions may develop from their initial conditions.
• Moreover, they allow for digitally manipulating, scaling, and examining graphs, providing a deeper insight into the behavior of mathematical functions.

Using CAS tools bridges the gap between abstract differential equations and concrete visual solutions, making them a vital resource in mathematics education.