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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 9.2.37b
Chapter 9, Problem 9.2.37b

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. Euler’s method is used to compute exact values of the solution of an initial value problem. 

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1
Recall that Euler's method is a numerical technique used to approximate solutions of initial value problems (IVPs) for differential equations, not to find exact solutions.
Understand that Euler's method works by taking small steps along the slope given by the differential equation, starting from the initial condition, to generate approximate values of the solution at discrete points.
Recognize that because Euler's method uses linear approximations over small intervals, the values it produces are approximations and generally contain some error compared to the exact solution.
Therefore, Euler's method does not compute exact values; instead, it provides an approximate solution that can be made more accurate by decreasing the step size.
Conclude that the statement 'Euler’s method is used to compute exact values of the solution of an initial value problem' is false, and the explanation is that Euler's method is inherently an approximation technique.

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Key Concepts

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

Euler’s Method

Euler’s method is a numerical technique used to approximate solutions of initial value problems for ordinary differential equations. It uses tangent line approximations at discrete steps to estimate the solution curve, rather than finding an exact formula.
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Euler's Method

Initial Value Problem (IVP)

An initial value problem consists of a differential equation along with a specified value of the unknown function at a given point. The goal is to find a function that satisfies both the differential equation and the initial condition.
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Initial Value Problems

Exact vs. Approximate Solutions

Exact solutions satisfy the differential equation and initial conditions precisely, often expressed in closed-form formulas. Approximate solutions, like those from Euler’s method, provide numerical estimates that approach the exact solution as the step size decreases.
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Solutions to Basic Differential Equations
Related Practice
Textbook Question

Properties of stirred tank solutions


b. Verify that M(0) = M₀

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. The solution of a stirred tank initial value problem always approaches a constant as t→∞

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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.

b. Sketch the direction field, for t≥0.


y′(t) = 2y + 4

Textbook Question

{Use of Tech} Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t≥0. The relevant initial value problem is 

dM/dt=−rM ln(M/K),M(0)=M0, 

where r and K are positive constants and 0<M0<K.

b. Solve the initial value problem and graph the solution for r=1,K=4, and M0=1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor? 

Textbook Question

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.

b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.


x′(t) = 2x − xy, y′(t) = −y + xy

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Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.


b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.


y′(t) = 2t + 1, y(0) = 0; y(t) = t² + t

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