Problem 32 - Essential Calculus

Question

A differential equation, a point, and a slope field are given. A slope field consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the directions of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (To print an enlarged copy of the graph, go to the website www.mathgraphs.com.) (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). $$ \frac{d y}{d x}=e^{\sin x} \cos x, \quad(\pi, 2) $$

Step-by-Step Solution

Answer

First, the slope field is used to generate the graph of the approximate solution curves, one of which passes through the point (\( \pi\), 2). Then, the equation \( \frac{d y}{d x}=e^{\sin x} \cos x \) is integrated with help of the integrating factor. Upon substituting the initial condition, a particular solution is obtained which can be graphed and compared with the approximate solution. This four-step process gives an understanding of the differential equation and its solutions both graphically and algebraically.

Step 1: Sketching approximate solutions using the slope field

Using the slope field given by \( \frac{d y}{d x}=e^{\sin x} \cos x \), draw tangents at regular intervals. Then, smoothly connect these tangents to formulate curves in the slope field. One of these curves should pass through the point (\( \pi\), 2). This step will give a visual representation for the family of solutions to the given equation.

Step 2: Integrating the differential equation

To integrate the given differential equation, use the formula for integrating factors, where the integrating factor of the differential equation would be \(e^{\int \sin(x) \, dx}\). Multiply both sides of the given equation by this factor and integrate. Then, simplify the equation and solve for \(y\).

Step 3: Plotting the particular solution

After finding the general solution from step two, now plug in the initial condition point (\( \pi\), 2) into the equation to solve for the constant. Then use a graphing calculator to sketch this solution curve, which represents the particular solution to the differential equation passing through this initial condition.

Step 4: Comparing the result with the sketches

Compare the sketch of the curve obtained in step 3, with the approximate solution curves derived in step 1. This would provide a visual confirmation of the legitimacy of the integration carried out in step 2 on the differential equation.

Key Concepts

Slope Fields

Slope fields, sometimes called direction fields, offer a way to visualize the solutions of a differential equation without actually solving it. Imagine a field of short line segments or tiny arrows drawn at grid points in the plane. Each segment has a slope determined by the differential equation. Here, the equation is \( \frac{d y}{d x} = e^{\sin x} \cos x \). To construct a slope field, compute the slope for various \(x\) values, then draw small line segments representing these slopes.Visualizing slope fields helps in understanding how solutions behave. They suggest the trajectories solutions might take. To sketch a solution:

• Locate the starting point, in this case, \((\pi, 2)\).
• Draw a curve following the direction indicated by the line segments.
• Continue the curve smoothly through the field, keeping tangent to the lines.

This way, we obtain a rough sketch of potential solution curves of the differential equation.

Particular Solutions

Finding particular solutions involves determining the exact solution to a differential equation that passes through a given point. For the given equation, the point is \((\pi, 2)\). Particular solutions are crucial as they provide specific solutions from the family of general solutions.To find it:

• First, solve the general solution of the differential equation by integrating.
• Plug the given point \((\pi, 2)\) into the general solution to find the constant of integration.

This method ties the solution to the specific conditions of the problem. You get a unique solution curve that matches the visual representation you created with the slope field.

Integration Techniques

Integration is key to solving differential equations and finding the general solution. The equation \( \frac{d y}{d x} = e^{\sin x} \cos x \) means we need to integrate \( e^{\sin x} \cos x \) to find \(y\).The technique involves using integrating factors or sometimes substitution methods.

• Identify the expression inside the integral.
• Choose the appropriate method for integration. Here, it might involve substitution due to the nature of sine and cosine functions.
• Simplify and solve the integral to find the general form of \( y \).

These steps require familiarity with various integration methods, allowing you to tackle different forms of differential equations.

Graphing Utilities

To confirm solutions and visualize the differential equation's outcome, graphing utilities are indispensable. They help draw precise solution curves, verify hand-drawn sketches, and explore the behavior of solutions over broader ranges.Using a graphing calculator or software:

• Input the general solution equation and the particular condition \((\pi, 2)\).
• Plot the curve to see the specific solution line accurately.
• Compare this plot with the earlier hand-made sketches to verify accuracy.

By graphing, one can quickly and visually verify that the integrated, steady-state solution aligns with the intuitive direction given by the slope field, ensuring a deeper understanding and validation of the process.