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Chapter 9: Problem 28

(a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b) Solve the differential equation. (c) Use the CAS to draw several members of the family of solutions obtained in part (b). Compare with the curves from part (a). \( y' = xy \)

Short Answer

Expert verified
Direction field and solved differential equation align; solutions are \( y = Ce^{\frac{x^2}{2}} \).

Step by step solution

01

Direction Field

Begin by using a computer algebra system (CAS) such as Mathematica, MATLAB, or Python (with libraries like Matplotlib) to plot the direction field for the differential equation \( y' = xy \). The direction field visually represents the slope of the tangent to the solution curves at any given point \((x, y)\). In CAS, set up a grid of points over a window of interest, compute the slope \( y' = xy \), and draw small linear segments to represent these slopes. After obtaining the direction field, sketch some potential solution curves that follow the slope lines.
02

Solve the Differential Equation

The differential equation \( y' = xy \) is separable. Separate the variables by rewriting the equation as \( \frac{dy}{y} = x \, dx \). Integrate both sides: \( \int \frac{1}{y} \, dy = \int x \, dx \). This leads to \( \ln|y| = \frac{x^2}{2} + C \), where \( C \) is the integration constant. Exponentiate both sides to solve for \( y \): \( y = \pm e^{C} e^{\frac{x^2}{2}} = Ce^{\frac{x^2}{2}} \), where \( C \) can be any non-zero constant (redefining \( C \) to absorb the sign and exponential constant).
03

Plotting Solution Curves

Use the CAS again to plot several members of the family of solutions \( y = Ce^{\frac{x^2}{2}} \). Choose different values for the constant \( C \) to obtain and display multiple curves. Overlay these solution curves on the direction field obtained in Step 1 to verify that they align with the direction indicated by the arrows in the field. Each solution curve should follow the direction of the field lines at every point.
04

Compare Solution Curves with Direction Field

Compare the solution curves plotted in Step 3 with the sketch from Step 1. Ideally, the curves will align with the general movement suggested by the direction field lines drawn initially. This indicates that the analytical solution matches the visual slope patterns represented in the direction field.

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Key Concepts

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

Understanding Direction Fields
A direction field is a graphical tool used to visualize differential equations without explicitly solving them. For a given differential equation, such as \( y' = xy \), the direction field represents the slopes of the solution curves at various points on a plane. This is achieved by computing the slope \( y' = xy \) at different points \((x, y)\) and drawing small line segments to indicate the direction of the tangent at these points.
  • Each segment is a tiny arrow showing the direction in which the solution curve would travel if it passed through that point.
  • Direction fields help provide an intuition of the behavior of solutions over an area, offering a visual representation of the various possible trajectories of solutions.

Tools like computer algebra systems make generating these fields easier by automating the calculations and graphs. Once the field is generated, you can attempt to sketch solution curves that follow the general direction of these arrows.
Exploring Separable Equations
A differential equation is termed 'separable' if it can be rearranged to express the variables independently on opposite sides of the equation. The given equation \( y' = xy \) is separable, allowing us to rearrange and solve it more easily.
  • We start by expressing it as \( \frac{dy}{y} = x \, dx \), effectively segregating the variable \(y\) on one side and \( x \) on the other.
  • Separable equations can be solved by direct integration once variables are separated. Thus, integrating both sides gives \( \ln|y| = \frac{x^2}{2} + C \).

Separable equations are a crucial set of differential equations because their solutions are straightforward, involving basic integration skills. This makes them a fundamental concept when studying differential equations.
Using a Computer Algebra System
Computer algebra systems (CAS) such as Mathematica, MATLAB, or Python with libraries like Matplotlib are powerful tools for solving differential equations and visualizing their solutions. These systems can automatically compute direction fields and solution curves, making complex equations more manageable.
  • They handle calculations and graph generation efficiently, saving time and reducing human error.
  • A CAS can help in plotting a direction field by automating the slope calculations at numerous points.

Moreover, once we have a family of solution curves, a CAS can easily plot these curves, helping to verify analytical solutions with visual representations. This overlap assures that the solutions are consistent with the behavior denoted by the direction field.
Interpreting Solution Curves
Solution curves are specific paths that satisfy a differential equation under given initial conditions or constants. For the equation \( y = Ce^{\frac{x^2}{2}} \), each curve corresponds to a unique constant value of \( C \).
  • These curves represent the range of possible solutions depending on the constant \( C \), enabling us to visualize different behavior patterns of the differential equation.
  • The curves should closely follow the direction indicated by the direction field arrows, further confirming the accuracy of the solution.

Seeing the solution curves laid over the direction field provides a complete picture of the system's dynamics. It bridges the gap between analytical solutions and their practical interpretations in a visual format. By comparing plotted solutions to direction field sketches, we establish a deeper comprehension of both individual solutions and the overall behavior standard indicated by the differential equation.

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Most popular questions from this chapter

A vat with 500 gallons of beer contains \( 4% \) alcohol (by volume). Beer with \( 6% \) alcohol is pumped into the vat at a rate of 5 gal/min and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour?

Suppose you have just poured a cup of freshly brewed coffee with temperature \( 95^{\circ} \) in a room where the temperature is \( 20^{\circ}. \) (a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain. (b) Newtons Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Write a differential equation that expresses Newtons Law of Cooling for this particular situation. What is the initial condition? In view of your answer to part (a), do you think this differential equation is an appropriate model for cooling? (c) Make a rough sketch of the graph of the solution of the initial-value problem in part (b).

Suppose that a population grows according to a logistic model with carrying capacity 6000 and \( k = 0.0015 \) per year. (a) Write the logistic differential equation for these data. (b) Draw a direction field (either by hand or with a computer algebra system). What does it tell you about the solution curves? (c) Use the direction field to sketch the solution curves for initial populations of 1000, 2000, 4000, and 8000. What can you say about the concavity of these curves? What is the significance of the inflection points? (d) Program a calculator or computer to use Euler's method with step size \( h = 1 \) to estimate the population after 50 years if the initial population is 1000. (e) If the initial populations is 1000, write a formula for the population after \( t \) years . Use it to find the population after 50 years and compare with your estimate in part (d). (f) Graph the solution in part (e) and compare with the solution curve you sketched in part (c).

Solve the initial-value problem. \( xy' = y + x^2 \sin x, y(\pi) = 0 \)

An object with mass \( m \) is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If \( s(t) \) is the distance dropped after \( t \) seconds, then the speed is \( v = s'(t) \) and the acceleration is \( a = v'(t). \) If \( g \) is the acceleration due to gravity, them the downward force on the object is \( mg - cv, \) where \( c \) is a positive constant, and Newton's Second Law gives \( m \frac {dv}{dt} = mg - cv \) (a) Solve this as a linear equation to show that \( v = \frac {mg}{c} (1 - e^{-ct/m}) \) (b) What is the limiting velocity? (c) Find the distance the object has fallen after \( t \) seconds.
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