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Chapter 8: Problem 18

Solve the differential equation and then use a graphing utility to generate five integral curves for the equation. $$ (\cos y) y^{\prime}=\cos x $$

Short Answer

Expert verified
Separate variables and integrate to find the implicit solution \( \ln |\sec y + \tan y| = \sin x + C \).

Step by step solution

01

Separate Variables

We begin by rearranging the differential equation to separate the variables. The given equation is \((\cos y) y^{\prime} = \cos x\).Rewriting with separation of variables, we have: \[ \frac{1}{\cos y} \, dy = \frac{\cos x}{1} \, dx. \]
02

Integrate Both Sides

Next, we integrate both sides of the equation separately.The integral on the left is \( \int \frac{1}{\cos y} \, dy = \int \sec y \, dy \), which becomes \( \ln |\sec y + \tan y| + C_1 \).The integral on the right is \( \int \cos x \, dx = \sin x + C_2 \).
03

Combine Results and Solve for y (Implicit Solution)

Combining the results from the integration, we have:\[ \ln |\sec y + \tan y| = \sin x + C, \] where \( C = C_2 - C_1 \).This represents an implicit solution of the differential equation.
04

Determine Specific Solutions (Integral Curves)

To find specific solutions, we choose different constants of integration, \( C \).Choosing arbitrary constants, such as \( C = 0, 1, -1, 2, -2 \), we acquire different integral curves.
05

Plot Integral Curves

Use a graphing utility to plot the integral curves for each chosen constant \( C \). The curves can be generated by substituting values into \( \ln |\sec y + \tan y| = \sin x + C \) and plotting the resulting functions.

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Key Concepts

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

Variable Separation
Variable separation is a technique used to make solving differential equations simpler. By separating the variables, we aim to express each variable on opposite sides of the equation. In our exercise, the original differential equation is \((\cos y) y^{\prime} = \cos x\). Our goal is to separate terms involving \(y\) from those involving \(x\).
Here's how to do it:
  • Move terms around to isolate one variable on each side.
  • Reframe the equation as \(\frac{1}{\cos y} \, dy = \cos x \, dx\).
Achieving variable separation sets us up for successful integration. With \(y\)-dependent terms on one side and \(x\)-dependent on the other, each can be integrated independently, propelling us forward in solving the equation.
Integration
Integration is crucial in solving the separated differential equation. It involves finding the antiderivative of each side separately. After separating the variables, we have \(\int \sec y \, dy = \int \cos x \, dx\). Integration proceeds as follows:
  • The left-hand integral becomes \(\ln |\sec y + \tan y| + C_1\). This requires knowledge of integration rules for trigonometric functions.
  • The right-hand integral evaluates to \(\sin x + C_2\), using the basic antiderivative of \(\cos x\).
Both results include constants of integration. By combining them, we describe a general implicit solution of the differential equation as \(\ln |\sec y + \tan y| = \sin x + C\). This implicit form is a key milestone towards understanding the behavior of the system described by the differential equation.
Integral Curves
Integral curves represent specific solutions to the differential equation, obtained by choosing different values for the constant \(C\). In our case, the solution \(\ln |\sec y + \tan y| = \sin x + C\) yields various curves based on the constant's value. Here's what to do:
  • Pick several values for \(C\), such as 0, 1, -1, 2, and -2.
  • Each value of \(C\) defines a unique curve, offering different possible states of the system.
Plotting these curves can visually demonstrate how solutions diverge based on initial conditions. Integral curves allow you to see the entire family of solutions, rather than just one isolated path.
Graphing Utility
Using a graphing utility aids in visualizing the integral curves derived from the differential equation. These tools can graph the various integral curves, providing insight into their shape and behavior. To ensure the graphing process is effective:
  • Substitute values of \(C\) from the implicit solution \(\ln |\sec y + \tan y| = \sin x + C\).
  • Enter these equations into a graphing calculator or software, like Desmos or GeoGebra.
  • Observe how changing \(C\) shifts and alters the curves.
Through graphing utilities, you gain a tangible understanding of how solutions to differential equations evolve, offering visual confirmation of theoretical calculations.

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Most popular questions from this chapter

Find a solution to the initial-value problem. $$ y^{\prime}+4 x=2, y(0)=3 $$

Suppose that a quantity \(y\) has an exponential growth model \(y=y_{0} e^{k t}\) or an exponential decay model \(y=y_{0} e^{-k t}\), and it is known that \(y=y_{1}\) if \(t=t_{1} .\) In each case find a formula for \(k\) in terms of \(y_{0}, y_{1}\), and \(t_{1}\), assuming that \(t_{1} \neq 0\).

True-False Determine whether the statement is true or false. Explain your answer. If the first-order linear differential equation $$ \frac{d y}{d x}+p(x) y=q(x) $$ has a solution that is a constant function, then \(q(x)\) is a constant multiple of \(p(x)\).

A scientist wants to determine the half-life of a certain radioactive substance. She determines that in exactly 5 days a \(10.0\) -milligram sample of the substance decays to \(3.5\) milligrams. Based on these data, what is the half-life?

(a) There is a trick, called the Rule of 70, that can be used to get a quick estimate of the doubling time or half-life of an exponential model. According to this rule, the doubling time or half-life is roughly 70 divided by the percentage growth or decay rate. For example, we showed in Example 5 that with a continued growth rate of \(1.10 \%\) per year the world population would double every 63 years. This result agrees with the Rule of 70 , since \(70 / 1.10 \approx 63.6\). Explain why this rule works. (b) Use the Rule of 70 to estimate the doubling time of a population that grows exponentially at a rate of \(1 \%\) per year. (c) Use the Rule of 70 to estimate the half-life of a population that decreases exponentially at a rate of \(3.5 \%\) per hour. (d) Use the Rule of 70 to estimate the growth rate that would be required for a population growing exponentially to double every 10 years.
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