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Chapter 9: Problem 70

The following exercises require the use of a slope field program. For each differential equation: a. Use a graphing calculator slope field program to graph the slope field for the differential equation on the window [-5,5] by [-5,5]. b. Sketch the slope field on a piece of paper and draw a solution curve that follows the slopes and that passes through the given point. $$ \begin{array}{l} \frac{d y}{d x}=x \ln \left(y^{2}+1\right) \\ \text { point: }(0,-2) \end{array} $$

Short Answer

Expert verified
Graph the slope field and draw the curve through (0, -2) using the slopes.

Step by step solution

01

Understanding the Differential Equation

The differential equation given is \( \frac{dy}{dx} = x \ln(y^2 + 1) \). Our task is to graph the slope field for this equation and draw a solution curve that passes through the point (0,-2). A slope field represents the tangent line (slope) at various points and helps visualize solutions.
02

Setting Up the Graphing Calculator

Use a graphing calculator or a slope field program. Set the graphing window to the range from -5 to 5 for both x and y axes. This creates a viewing window of [-5, 5] by [-5, 5]. Ensure the slope field function is set to \( \frac{dy}{dx} = x \ln(y^2 + 1) \).
03

Generating the Slope Field

Input the differential equation \( \frac{dy}{dx} = x \ln(y^2 + 1) \) into the calculator program. The program will calculate and display the slope at various (x, y) points within the specified window, thus generating the slope field.
04

Sketching the Slope Field

Sketch the slope field on paper by visualizing small line segments that represent the slopes, as calculated in the previous step, across the aforementioned grid [-5, 5] x [-5, 5]. This gives a visual representation of the differential equation.
05

Drawing the Solution Curve

Using the slopes as guidance, draw a curve on the paper that passes through the given initial point (0, -2). The curve should follow the direction of the slope lines around it, illustrating a possible solution to the differential equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical expressions that involve functions and their derivatives. In simpler terms, they describe how a certain quantity changes over time or space. When you see a differential equation like \( \frac{dy}{dx} = x \ln(y^2 + 1) \), it tells you how the variable \( y \) changes with respect to \( x \).
  • They often represent physical phenomena, such as the cooling of objects or population dynamics.
  • Solving a differential equation means finding a function \( y \) that satisfies this equation.
  • The solution gives insight into the behavior of the system described by the equation.
In our exercise, the differential equation defines a relationship between \( x \) and \( y \) through their derivatives, providing a foundation for plotting the slope field and understanding the solution curves.
Solution Curves
Solution curves are crucial for interpreting differential equations. They are the actual paths on a graph that represent solutions to the differential equation. These curves indicate how a particular system evolves.
  • Each solution curve is unique to its initial condition, such as the point \( (0,-2) \) in our exercise.
  • By sketching solution curves on a slope field, you can see potential behaviors of the system under various conditions.
  • These curves effectively illustrate how the function \( y \) behaves as \( x \) changes, giving a visual sense of direction and flow.
By drawing a solution curve through a specific point, we can pinpoint one particular solution from an entire family of possible solutions, which is particularly helpful in navigating complex systems.
Graphing Calculator
A graphing calculator or software is a powerful tool in visualizing differential equations and their solutions. These devices can quickly generate slope fields and other graphs by processing mathematical expressions like \( \frac{dy}{dx} = x \ln(y^2 + 1) \).
  • They help visualize the behavior of functions in a more dynamic way compared to paper-based methods.
  • Setting up a graphing window from -5 to 5 for both \( x \) and \( y \) axes ensures a comprehensive view of the slope field.
  • The calculator plots slope lines at various points, showing how steep or gentle the slopes are at those points.
By using a graphing calculator, students can enhance their understanding of differential equations by observing their graphically represented solutions, which is often easier to interpret and analyze.
Slope
The concept of slope is central to understanding differential equations and their graphical representations. In mathematics, the slope refers to the steepness or incline of a line, often described as the ratio of the "rise" over the "run" between two points.
  • In a slope field, the slope of each line segment is determined by the differential equation \( \frac{dy}{dx} \).
  • Slope fields consist of little line segments indicating the slope at various points in the \( xy \)-plane.
  • The direction and magnitude of these slopes help trace the solution curves of differential equations.
Understanding slopes is essential in predicting how a particular solution will evolve over time or space, making it an indispensable part of studying differential equations.
Tangent Line
Tangent lines are lines that touch a curve at a single point without crossing over, showing the direction of the curve at that point. In the context of differential equations, the slope of a tangent line at a point is given by the derivative \( \frac{dy}{dx} \).
  • These lines give immediate insight into the behavior of the curve at specific points.
  • In a slope field, tangent lines are represented by small line segments that approximate these behaviors across the plane.
  • When drawing solution curves, the direction of these tangents indicates the way the curve should flow through a particular region.
Using tangent lines, students can determine the immediate direction and behavior of a system modeled by a differential equation, leading to a better grasp of its overall dynamics.
Numerical Methods
Numerical methods are techniques often used to approximate solutions to differential equations that cannot be solved analytically. When dealing with complex equations, these methods provide ways to calculate values that help sketch solution curves.
  • Methods like Euler's method or Runge-Kutta rely on calculating successive points along a curve, essentially moving step-by-step.
  • These approaches are crucial when exact solutions are challenging or impossible to find.
  • Numerical methods complement the visual strategy of slope fields, offering another layer of understanding.
By applying numerical methods, students can approximate the behavior of the solution curve, allowing them to explore more intricate mathematical models that are otherwise elusive through purely analytical means.

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Most popular questions from this chapter

Determine whether each differential equation is separable. (Do not solve it, just find whether it's separable.) $$y^{\prime}=x+y$$

A Ponzi scheme is an investment fraud that promises high returns but in which funds, instead of being invested, are merely used to pay returns to the investors and profits to the fund manager. To avoid running out of money, new investors must be brought in at an increasing rate to provide funds to pay off existing investors. Eventually the scheme must collapse in debt when not enough new investors can be found. Suppose that each of 10 investors deposits \(\$ 100,000\) into a fund and is promised a \(20 \%\) annual return. However, the \(\$ 1,000,000\) collected is used to pay each investor the required \(\$ 20,000\) return, with the remaining \(\$ 800,000\) kept by the fund manager. Let \(y(t)\) be the total number of investors needed after \(t\) years so that incoming funds will be enough to pay the existing investors plus the manager's \(\$ 800,000 .\) Representing dollar amounts in thousands, we have $$ \begin{array}{l} \left(\begin{array}{c} \text { Annual } \\ \text { inflow } \end{array}\right)=100 \cdot \frac{d y}{d t} \\ \left(\begin{array}{l} \text { Annual } \\ \text { outflow } \end{array}\right)=20 \cdot y+800 \end{array} $$ For inflow to equal outflow, we must have the differential equation and initial condition $$ \left\\{\begin{array}{l} 100 \cdot \frac{d y}{d t}=20 y+800 \\ y(0)=10 \end{array}\right. $$ a. Solve this differential equation and initial condition. b. Use your solution to find how many investors would be needed after 10 years, after 20 years, after 30 years, and after 50 years.

For each initial value problem, use an Euler's method graphing calculator program to find the approximate solution at the stated \(x\) -value, using 50 segments. [Hint: Use an interval that begins at the initial \(x\) -value and ends at the stated \(x\) -value. \(y^{\prime}=e^{x / y}\) \(y(1)=0.4\) Approximate the solution at \(x=3\)

The burning of coal and oil is increasing the amount of carbon dioxide in the atmosphere, which is expected to trap more solar radiation and increase global temperatures (the "greenhouse effect"), possibly raising the sea level by melting polar ice and flooding coastal regions. The current 3300 billion tons of carbon dioxide in the atmosphere is growing by about 50 billion tons per year, while about \(1 \%\) of the accumulated carbon dioxide is removed by natural processes. Therefore, the amount \(y(t)\) of carbon dioxide (in billions of tons) \(t\) years from now satisfies $$ \begin{array}{l} y^{\prime}=-0.01 y+50 \\ y(0)=3300 \end{array} $$ a. Solve this differential equation and initial condition. b. Find when the amount of carbon dioxide will reach 4000 billion tons, at which time global temperatures are expected to have risen by 3 degrees Fahrenheit c. Use your solution to find the long-run amount of carbon dioxide in the atmosphere.

For the initial value problem \(\left\\{\begin{array}{l}y^{\prime}=4 x y \\\ y(1)=3\end{array}\right.\) state the initial point \(\left(x_{0}, y_{0}\right)\) and calculate the slope of the solution at this point.
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