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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.3.54c
Chapter 9, Problem 9.3.54c

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.
c. Graph the solutions in part (b) and describe their behavior as t increases. 

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Recall the differential equation given: \(y \cdot y'(t) = \frac{1}{2} e^{t} + t\). From part (b), you should have found the general solution \(y(t)\) by separating variables and integrating both sides.
To graph the solutions, first express \(y'(t)\) explicitly by dividing both sides by \(y\): \(y'(t) = \frac{\frac{1}{2} e^{t} + t}{y}\). However, since you have the implicit or explicit form of \(y(t)\) from part (b), use that expression to plot \(y\) against \(t\).
Use a graphing tool or software to plot the solution curves for various initial conditions. This will help visualize how \(y(t)\) behaves as \(t\) increases.
Analyze the behavior of the solutions as \(t \to \infty\). Consider the dominant terms in the solution and how the exponential \(e^{t}\) and the linear term \(t\) influence the growth or decay of \(y(t)\).
Describe the qualitative behavior: Does \(y(t)\) grow without bound, approach a finite limit, or oscillate? Note how different initial conditions affect the long-term behavior of the solutions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Separable Differential Equations

A separable differential equation can be written so that all terms involving the dependent variable y are on one side and all terms involving the independent variable t are on the other. This allows integration of both sides separately to find the general solution. Recognizing and manipulating the equation into this form is essential for solving it.
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Solving Separable Differential Equations

Behavior of Solutions and Graphing

Analyzing the behavior of solutions involves understanding how the function y(t) changes as t increases, including growth, decay, or approaching asymptotes. Graphing solutions helps visualize these trends and identify long-term behavior, stability, or oscillations, which is crucial for interpreting the differential equation's implications.
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Graphing The Derivative

Initial Conditions and Particular Solutions

Initial conditions specify the value of the solution at a particular point, allowing determination of a unique particular solution from the general family. This is important for graphing specific solution curves and accurately describing their behavior over time, as different initial values can lead to different solution trajectories.
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Initial Value Problems
Related Practice
Textbook Question

Another second-order equation Consider the differential equation y''(t) + k²y(t) = 0, where k is a positive real number.

c. Give the general solution of the equation for arbitrary k > 0 and verify your conjecture.

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Textbook Question

Solving Bernoulli equations Use the method outlined in Exercise 43 to solve the following Bernoulli equations.


c. y′(t) + y = √y

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Textbook Question

{Use of Tech} Fish harvesting A fish hatchery has 500 fish at t=0, when harvesting begins at a rate of b>0fish/year The fish population is modeled by the initial value problem y′(t)=0.01y−b,y(0)=500 where t is measured in years.


c. Graph the solution in the case that b=60fish/year. Describe the solution.

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Textbook Question

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.


c. Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing?


y'(t) = cos y for |y| ≤ π

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Textbook Question

{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation 


m · v'(t) = mg + f(v)

mass | acceleration | external forces


where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.


c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a. 

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Textbook Question

Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.

c. Draw a representative direction field in the case that a<0. Show that if A>−b/a, then the solution decreases for t≥0, and that if A<−b/a, then the solution increases for t≥0.

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