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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 9.2.30d
Chapter 9, Problem 9.2.30d

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


d. In general, how does halving the time step affect the error at t=0.2 and t=0.4?


y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

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Recall that Euler's method approximates the solution to the differential equation by using the formula \(y_{n+1} = y_n + h f(t_n, y_n)\), where \(h\) is the time step size.
Understand that the local truncation error of Euler's method at each step is proportional to \(h^2\), and the global error (error accumulated over multiple steps) is proportional to \(h\).
Since the global error is roughly proportional to the step size \(h\), halving the time step \(h\) will approximately halve the error at any fixed time \(t\), such as \(t=0.2\) and \(t=0.4\).
To see this concretely, consider the number of steps needed to reach \(t=0.2\) and \(t=0.4\) with step size \(h\) and with step size \(h/2\). The smaller step size means more steps but smaller error per step, resulting in an overall smaller total error.
Therefore, halving the time step reduces the global error roughly by a factor of 2 at the specified times, improving the accuracy of Euler's method for this problem.

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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 to approximate solutions of initial value problems for ordinary differential equations. It uses a stepwise approach, estimating the next value by moving along the slope (derivative) at the current point multiplied by a small time step. The accuracy depends on the step size chosen.
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Euler's Method

Local and Global Truncation Error

Local truncation error is the error made in a single step of Euler's method, while global truncation error accumulates over multiple steps. For Euler's method, the local error is proportional to the square of the step size, and the global error is proportional to the step size, meaning smaller steps reduce overall error.
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Determining Error and Relative Error

Effect of Step Size on Error

Halving the time step in Euler's method generally reduces the global error approximately by half, improving accuracy. This is because the global error is linearly dependent on the step size, so smaller steps lead to more precise approximations at given points like t=0.2 and t=0.4.
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Determining Error and Relative Error
Related Practice
Textbook Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.


where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.


d. Find lim(t→∞) P(t) and check that the result is consistent with the graph in part (c).

Textbook Question

U.S. population projections According to the U.S. Census Bureau, the nation’s population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach:


d. Estimations of this kind must be made and interpreted carefully. Suppose the projected population for 2050 is 410 million rather than 398 million. What is the value of the carrying capacity in this case?

Textbook Question

A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time t = 0, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant.

d. After how many minutes does the drug mass reach 90% of its steady-state level? 

Textbook Question

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

d. Compare the errors in the approximations to y(T).


y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

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Textbook Question

[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. 

Textbook Question

A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.


d. For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C₁cosh(kt) + C₂sinh(kt), where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively (Section 7.3).