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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 9.1.49a
Chapter 9, Problem 9.1.49a

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The general solution of the differential equation y'(t) = 1 is y(t) = t

Verified step by step guidance
1
Identify the given differential equation: \(y'(t) = 1\). This means the derivative of \(y\) with respect to \(t\) is constantly 1.
Recall that the general solution to a differential equation of the form \(y'(t) = c\), where \(c\) is a constant, is \(y(t) = ct + C\), where \(C\) is an arbitrary constant of integration.
Apply this to the given equation: since \(c = 1\), the general solution should be \(y(t) = 1 \cdot t + C\), or simply \(y(t) = t + C\).
Compare this with the proposed solution \(y(t) = t\). Notice that the proposed solution does not include the constant of integration \(C\).
Conclude that the statement is false because the general solution must include the constant \(C\) to account for all possible solutions, not just \(y(t) = t\).

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Key Concepts

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

General Solution of a Differential Equation

The general solution of a differential equation includes all possible solutions and typically contains an arbitrary constant. It represents a family of functions that satisfy the equation, not just a single function.
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Integration of Derivatives

To find the general solution of y'(t) = 1, integrate the right-hand side with respect to t. Since the derivative of y is 1, integrating gives y(t) = t + C, where C is an arbitrary constant.
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Counterexample and Verification

To verify if a proposed solution is general, check if it includes all solutions. The statement y(t) = t omits the constant C, so it is not the general solution. A counterexample is y(t) = t + 5, which also satisfies y'(t) = 1.
Related Practice
Textbook Question

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.

a. Find the general solution of the equation and express it explicitly as a function of t in two cases: y > 0 and y < 0.

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Textbook Question

{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm. 

a. Find the solution of the initial value problem.

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Textbook Question

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


a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).


y'(t) = (y−2)(y+1)

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Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].


y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ

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Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions.


y′(t) = 2y + 4

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Textbook Question

46–48. Analyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields.

Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′(t)+km(t)=I, where m(t) is the mass of the drug in the blood at time t≥0, K is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. Let I=10mg/hr and k=0.05 hr^−1.

a. Draw the direction field, for 0≤t≤100, 0≤y≤600.

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