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Chapter 7: Problem 7

Consider the initial value problem $$ \frac{d y}{d t}=-\frac{t}{y}, y(0)=8 $$ a. Find the solution of the initial value problem and sketch its graph. b. For what values of \(t\) is the solution defined? c. What is the value of \(y\) at the last time that the solution is defined? d. By looking at the differential equation, explain why we should not expect to find solutions with the value of \(y\) you noted in (c).

Short Answer

Expert verified
The solution is \(y = 8e^{-t^2/2}\), defined for all real \(t\). The smallest value of \(y\) is 0 as \(t\) approaches 卤鈭�, but such a solution cannot actually exist.

Step by step solution

01

- Separate the Variables

Start by separating the variables: Move all terms involving y to one side and all terms involving t to the other side: \[\frac{dy}{y} = -t \, dt\]
02

- Integrate Both Sides

Integrate both sides of the equation:\[\int \frac{1}{y} \, dy = \int -t \, dt\] This gives: \[\ln |y| = -\frac{t^2}{2} + C\]
03

- Solve for the Constant C Using the Initial Condition

Use the initial condition y(0) = 8 to find C. Substitute t = 0 and y = 8: \[\ln 8 = C\] Thus, \[C = \ln 8\]
04

- Solve for y

Rewrite the equation to solve for y: \[\ln |y| = -\frac{t^2}{2} + \ln 8\] Exponentiate both sides to solve for y: \[|y| = e^{\ln 8 - \frac{t^2}{2}} = 8e^{-t^2/2}\] Since y is positive, \[y = 8e^{-t^2/2}\]
05

- Determine the Domain

The solution \(y = 8e^{-t^2/2}\) is defined for all real \(t\) because the exponential function is defined for all real numbers.
06

- Last Time Value

Observe that the exponential function \(8e^{-t^2/2}\) approaches 0 as \(t\) approaches 卤鈭�. Thus, the smallest value \(y\) can attain is 0, but this occurs as \(t\) approaches 卤鈭�, not at any finite value.
07

- Differential Equation Explanation

By examining \(\frac{dy}{dt} = -\frac{t}{y}\), as \(y\) approaches 0, the right-hand side becomes undefined because division by zero is not possible. Therefore, a solution with \(y = 0\) cannot exist.

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Key Concepts

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

Differential Equations
A Differential Equation is an equation that involves the derivatives of a function. In this problem, we have the differential equation \(\frac{d y}{d t}=-\frac{t}{y} \). Such equations describe how a function changes and are central in many fields of science and engineering. To solve these equations, we need to find a function that satisfies the equation. Here, our goal is to find a function y(t) that meets the given differential equation and initial condition.
Separation of Variables
Separation of Variables is a method used to solve differential equations. It's particularly useful when the variables can be separated on opposite sides of the equation. In this exercise, we start by separating the variables: \(\frac{dy}{y} = -t \, dt \). By moving all terms involving y to one side and all terms involving t to the other side, we simplify the equation and prepare it for integration. This method is powerful because it reduces a complex problem into simpler, integrable parts.
Exponential Function
The Exponential Function appears frequently in differential equations solutions. In this problem, when we solved for y, we ended up with \(|y| = 8e^{-t^2/2} \). Exponentiating both sides allows us to solve the equation involving a natural logarithm (ln). The exponential function has unique properties: its rate of change is proportional to its value, and it's defined for all real numbers. These make it crucial in describing growth and decay processes.
Integration
Integration is the process of finding the antiderivative of a function. After separating the variables, we integrate both sides: \(\begin{aligned} \int \frac{1}{y} \, dy &= \int -t \, dt \end{aligned} \). This gives us the function \(\begin{aligned} \ln |y| &= -\frac{t^2}{2} + C \end{aligned} \). Integration is essential here because it allows us to solve for y in terms of t. Then, we use the initial condition to determine the constant C.
Initial Condition
The Initial Condition is a necessary part of solving an initial value problem. It gives us a specific point on the function's graph. In this problem, the initial condition is given by \y(0) = 8\. We use this to find the constant of integration (C). Plugging in the values, we get \(\begin{aligned} \ln 8 &= C \end{aligned} \). This information allows us to complete the solution by giving us a starting point. Without the initial condition, we'd have a general solution rather than a specific one.

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Most popular questions from this chapter

The table below gives the percentage, \(P\), of households with a VCR, as a function of year. \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline Year & 1978 & 1979 & 1980 & 1981 & 1982 & 1983 & 1984 \\ \hline P & 0.3 & 0.5 & 1.1 & 1.8 & 3.1 & 5.5 & 10.6 \\ \hline Year & 1985 & 1986 & 1987 & 1988 & 1989 & 1990 & 1991 \\ \hline P & 20.8 & 36.0 & 48.7 & 58.0 & 64.6 & 71.9 & 71.9 \\ \hline \end{tabular} (a) A logistic model is a good one to use for these data. Explain why this might be the case: logically, how large would the growth in VCR ownership be when they are first introduced? How large can the ownership ever be? We can also investigate this by estimating the growth rate of \(P\) for the given data. Do this at the beginning, middle, and near the end of the data: \(P^{\prime}(1980) \approx\) \(P^{\prime}(1985) \approx\) \(P^{\prime}(1990) \approx\) Be sure you can explain why this suggests that a logistic model is appropriate. (b) Use the data to estimate the year when the point of inflection of \(P\) occurs. The inflection point occurs approximately at (Give the year in which it occurs.) What percent of households had VCRs then? \(P=\) What limiting value \(L\) does this point of inflection predict (note that if the logistic model is reasonable, this prediction should agree with the data for 1990 and 1991 )? \(L=\) (c) The best logistic equation (solution to the logistic differential equation) for these data turns out to be the following. $$ P=\frac{75}{1+316.75 e^{-0.699 t}} . $$ What limiting value does this predict? $$ L= $$

Suppose that a cylindrical water tank with a hole in the bottom is filled with water. The water, of course, will leak out and the height of the water will decrease. Let \(h(t)\) denote the height of the water. A physical principle called Torricelli's Law implies that the height decreases at a rate proportional to the square root of the height. a. Express this fact using \(k\) as the constant of proportionality. b. Suppose you have two tanks, one with \(k=-1\) and another with \(k=-10 .\) What physical differences would you expect to find? c. Suppose you have a tank for which the height decreases at 20 inches per minute when the water is filled to a depth of 100 inches. Find the value of \(k\). d. Solve the initial value problem for the tank in part (c), and graph the solution you determine. e. How long does it take for the water to run out of the tank? f. Is the solution that you found valid for all time \(t ?\) If so, explain how you know this. If not, explain why not.

The population of a species of fish in a lake is \(P(t)\) where \(P\) is measured in thousands of fish and \(t\) is measured in months. The growth of the population is described by the differential equation $$ \frac{d P}{d t}=f(P)=P(6-P) $$ a. Sketch a graph of \(f(P)=P(6-P)\) and use it to determine the equilibrium solutions and whether they are stable or unstable. Write a complete sentence that describes the long-term behavior of the fish population. b. Suppose now that the owners of the lake allow fishers to remove 1000 fish from the lake every month (remember that \(P(t)\) is measured in thousands of fish). Modify the differential equation to take this into account. Sketch the new graph of \(d P / d t\) versus P. Determine the new equilibrium solutions and decide whether they are stable or unstable. c. Given the situation in part (b), give a description of the long-term behavior of the fish population. d. Suppose that fishermen remove \(h\) thousand fish per month. How is the differential equation modified? e. What is the largest number of fish that can be removed per month without eliminating the fish population? If fish are removed at this maximum rate, what is the eventual population of fish?

During the first few years of life, the rate at which a baby gains weight is proportional to the reciprocal of its weight. a. Express this fact as a differential equation. b. Suppose that a baby weighs 8 pounds at birth and 9 pounds one month later. How much will he weigh at one year? c. Do you think this is a realistic model for a long time?

An unknown radioactive element decays into non-radioactive substances. In 320 days the radioactivity of a sample decreases by 58 percent. (a) What is the half-life of the element? half-life: (b) How long will it take for a sample of \(100 \mathrm{mg}\) to decay to \(88 \mathrm{mg} ?\) time needed:
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