Problem 32 - Thomas Calculus

Question

In Exercises 31 and \(32,\) obtain a slope field and graph the particular solution over the specified interval. Use your CAS DE solver to find the general solution of the differential equation. $$ y^{\prime}=(\sin x)(\sin y), \quad y(0)=2 ; \quad-6 \leq x \leq 6, \quad-6 \leq y \leq 6 $$

Step-by-Step Solution

Answer

Find the slope field, use a CAS to solve, and apply the initial condition to get the particular solution.

Step 1: Understand the Differential Equation

The given differential equation is \( y' = \sin(x) \sin(y) \). We need to analyze this equation to find solutions and better understand how solutions behave for different initial conditions.

Step 2: Draw the Slope Field

Plot small line segments at various points in the \(xy\)-plane, where the slope of each segment is given by \( \sin(x) \sin(y) \). This visual representation will help us understand the direction field and behavior of potential solutions through the plane from \(-6 \leq x \leq 6\) and \(-6 \leq y \leq 6\).

Step 3: Solve the Differential Equation using CAS

To find the general solution of the differential equation, utilize a Computer Algebra System (CAS) DE solver. The DE solver will integrate the equation \( y' = \sin(x)\sin(y) \) to find an expression for \(y\) as a function of \(x\).

Step 4: Apply Initial Condition

Use the initial condition \( y(0) = 2 \) to determine the particular solution. Substitute \( x = 0 \) and \( y = 2 \) into the general solution obtained from the CAS and solve for any integration constants.

Step 5: Graph the Particular Solution

Using the particular solution found with the initial condition, graph this solution over the specified interval \(-6 \leq x \leq 6\). Compare and ensure that your graph fits within the slope field plotted earlier.

Key Concepts

Slope Field

A slope field, also known as a direction field, is a visual representation of a differential equation in the xy-plane. It is composed of small line segments whose slopes at specific points correspond to the value of \( y' = \sin(x) \sin(y) \). The purpose of creating a slope field is to visualize how the solutions of a differential equation may behave as they progress through various points in the plane.

• Each line segment on the slope field indicates the direction or "slope" of the solution at that point.
• By observing the pattern of these segments, you can infer the behavior of solutions without solving the equation explicitly.
• A slope field provides an intuitive grasp of the dynamics dictated by the differential equation, especially useful for nonlinear equations like the one we are investigating.

To construct a slope field for the equation \( y' = \sin(x) \sin(y) \), plot line segments with slopes equal to \( \sin(x) \sin(y) \) at various points within the intervals \(-6 \leq x \leq 6\) and \(-6 \leq y \leq 6\).
This approach helps to visually predict how solutions might travel in different regions of the plane.

Initial Condition

An initial condition in differential equations is a specific value that a solution must satisfy at a certain point. In the given problem, the initial condition is \( y(0) = 2 \). This initial value is crucial since it allows us to determine one unique particular solution from the family of solutions described by the general solution.

• Initial conditions are used to pin down the constants in the general solution when you substitute them into the equation.
• In our case, \, having the initial condition \( y(0) = 2 \), implies we use \( x = 0 \) and \( y = 2 \) to solve for any integration constants.
• This process turns the general solution into a particular one, serving the specific requirement when \( x = 0 \), the solution should yield the result \( y = 2 \).

Initial conditions are essential in outlining how a particular solution emanates from a specified point, thereby concertizing an entire trajectory that a solution will follow.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a highly valuable tool in solving complex differential equations. CAS software such as Maple, Mathematica, or MATLAB can handle symbolic manipulations, making the process of solving differential equations more efficient, especially when dealing with cumbersome calculations or nonlinear equations like our current task.

• Using a CAS, you can input the differential equation \( y' = \sin(x) \sin(y) \) into the solver, and it will compute the general solution for you.
• The system integrates the equation symbolically, providing a functional relationship \( y = f(x) + C \), where \( C \) represents constants of integration.
• Further, CAS can handle complex algebraic steps that would be tedious if performed by hand, saving time and reducing errors.

Once the solution is generated by CAS, it can seamlessly guide the next stages in the problem. This includes applying initial conditions and visualizing the particular solution.

Graphing Solutions

Graphing solutions to differential equations allows for a clear understanding of how solutions behave over a designated range. After deriving the particular solution using a CAS and applying the initial condition, plotting this solution on a graph provides a visual comparison with the pre-drawn slope field.

• The graph of the particular solution should fit within the confines of the slope field, showcasing the solution's trajectory through the slope segments outlined earlier.
• Using graphical tools, either manually or via software, provides immediate insight into the accuracy and appropriateness of your particular solution.
• If done correctly, you will observe that the graphed solution follows the directional flow indicated by the slope field, crossing through the point where the initial condition was applied.

Graphing not only confirms the validity of the mathematical solution but also reinforces conceptual understanding of how differential equations describe changes and patterns dynamically.