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Chapter 10: Problem 4

Suppose that \(f(t)\) satisfies the initial-value problem \(y^{\prime}=y^{2}+t y-7, y(0)=2\). Is the graph of \(f(t)\) increasing or decreasing at \(t=0\) ?

Short Answer

Expert verified
The graph of \(f(t)\) is decreasing at \(t=0\).

Step by step solution

01

- Understand the Problem

Given the differential equation \(y^{\prime}=y^{2}+ty-7\) and the initial condition \(y(0)=2\), determine whether the graph of \(f(t)\), which solves this initial-value problem, is increasing or decreasing at \(t=0\).
02

- Substitute Initial Condition

Substitute \(t=0\) and \(y(0)=2\) into the differential equation. This gives \(y^{\prime}(0)=2^{2}+0\cdot2-7\).
03

- Simplify the Expression

Simplify the right side: \(2^{2}+0\cdot2-7 = 4 - 7 = -3\). Thus, \(y^{\prime}(0) = -3\).
04

- Determine Increasing or Decreasing

Since \(y^{\prime}(0) = -3 < 0\), this means the graph of \(f(t)\) is decreasing at \(t=0\).

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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 involves an unknown function and its derivatives. The goal is to find this function. In our problem, the differential equation is given by \( y' = y^2 + ty - 7 \). This means that the rate of change of \( y \) (denoted by \( y' \)) depends on both \( y \), and \( t \). To solve it, we seek a function \( y(t) \) that satisfies this relationship for any \( t \).
Differential equations can describe a wide range of physical, biological, and economic systems. When you read \( y' = y^2 + ty - 7 \), think about how the value of \( y \) changes depending on itself and \( t \). The solution to such an equation isn鈥檛 just a simple function鈥� it often involves complex behaviors based on the interactions of \( y \) and \( t \).
Initial Conditions
Initial conditions specify the value of the unknown function at a particular point, usually when \( t = 0 \). These conditions are crucial because they allow us to find the unique solution to a differential equation out of potentially many possibilities. In our problem, the initial condition is \( y(0) = 2 \).
Initial conditions ensure the correct 'starting point' for the solution. Think of it as fixing one end of a string before pulling it tight across various points. No matter how many different ways you pull the string, it will always start from that fixed point, which in this context is \( (0, 2) \). Without this, we cannot definitively determine the behavior of the function.
Increasing and Decreasing Functions
The terms 'increasing' and 'decreasing' describe how a function behaves with changing \( t \). An increasing function means \( f(t) \) gets larger as \( t \) increases, while a decreasing function means \( f(t) \) gets smaller. To determine this for a differential equation, we look at the sign of the derivative \( y' \).
In our solution, after substituting the initial condition \( y(0) = 2 \) into the differential equation, we found \( y'(0) = -3 \). This negative value indicates that the function \( f(t) \) is decreasing at \( t = 0 \). Understanding this helps predict the short-term behavior of the function around \( t = 0 \), which is very useful in practical applications. For instance, in population dynamics, an initial decreasing growth rate might suggest a declining population at the start.

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Most popular questions from this chapter

A body was found in a room when the room's temperature was \(70^{\circ} \mathrm{F}\). Let \(f(t)\) denote the temperature of the body \(t\) hours from the time of death. According to Newton's law of cooling, \(f\) satisfies a differential equation of the form $$ y^{\prime}=k(T-y) $$ (a) Find \(T\). (b) After several measurements of the body's temperature, it was determined that when the temperature of the body was 80 degrees it was decreasing at the rate of 5 degrees per hour. Find \(k\). (c) Suppose that at the time of death the body's temperature was about normal, say \(98^{\circ} \mathrm{F}\). Determine \(f(t)\). (d) When the body was discovered, its temperature was \(85^{\circ} \mathrm{F}\). Determine how long ago the person died.

Solve the following differential equations: $$ y^{\prime}=\left(\frac{t}{y}\right)^{2} e^{t^{3}} $$

In an autocatalytic reaction, one substance is converted into a second substance in such a way that the second substance catalyzes its own formation. This is the process by which trypsinogen is converted into the enzyme trypsin. The reaction starts only in the presence of some trypsin, and each molecule of trypsinogen yields one molecule of trypsin. The rate of formation of trypsin is proportional to the product of the amounts of the two substances present. Set up the differential equation that is satisfied by \(y=f(t)\), the amount (number of molecules) of trypsin present at time \(t .\) Sketch the solution. For what value of \(y\) is the reaction proceeding the fastest? [Note: Letting \(M\) be the total amount of the two substances, the amount of trypsinogen present at time \(t\) is \(M-f(t) .]\)

A person took out a loan of $$\$ 100,000$$ from a bank that charges \(7.5 \%\) interest compounded continuously. What should be the annual rate of payments if the loan is to be paid in full in exactly 10 years? (Assume that the payments are made continuously throughout the year.)

A Savings Account A person deposits $$\$ 10,000$$ in a bank account and decides to make additional deposits at the rate of \(A\) dollars per year. The bank compounds interest continuously at the annual rate of \(6 \%\), and the deposits are made continuously into the account. (a) Set up a differential equation that is satisfied by the amount \(f(t)\) in the account at time \(\bar{t}\). (b) Determine \(f(t)\) (as a function of \(A\) ). (c) Determine \(A\) if the initial deposit is to double in 5 years.
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