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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 9.2.34d
Chapter 9, Problem 9.2.34d

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

Verified step by step guidance
1
Identify the differential equation and initial condition: \(y'(t) = 6 - 2y\), with \(y(0) = -1\).
Set the step size \(\Delta t = 0.2\) and the target time \(T = 3\). Calculate the number of steps as \(n = \frac{T}{\Delta t} = \frac{3}{0.2} = 15\) steps.
Use Euler's method formula to approximate \(y\) at each step: \(y_{k+1} = y_k + \Delta t \cdot f(t_k, y_k)\), where \(f(t, y) = 6 - 2y\). Start with \(y_0 = -1\) and iterate up to \(k = 15\).
Calculate the exact solution at \(T=3\) using the given formula: \(y(3) = 3 - 4e^{-2 \cdot 3} = 3 - 4e^{-6}\).
Find the error by subtracting the Euler approximation at \(T=3\) from the exact solution: \(\text{Error} = |y(3) - y_{15}|\). This will allow you to compare the accuracy of the Euler method with the exact value.

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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 first-order differential equations. It uses a step size Δt to iteratively estimate the value of y at successive points by applying the formula y_{n+1} = y_n + Δt * f(t_n, y_n). This method is especially useful when an exact solution is difficult to obtain.
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Euler's Method

Error Analysis in Numerical Methods

Error analysis involves comparing the approximate numerical solution to the exact solution to measure accuracy. The error at a point is the difference between the exact value y(T) and the Euler approximation. Understanding how step size affects error helps improve approximation quality.
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Determining Error and Relative Error

Analytical Solution of Differential Equations

The analytical solution provides the exact formula for y(t), here given as y(t) = 3 - 4e^{-2t}. It serves as a benchmark to evaluate numerical methods. Knowing the exact solution allows direct computation of y(T) for error comparison with Euler's approximation.
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Solutions to Basic Differential Equations
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

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


d. According to Newton’s Law of Cooling, the temperature of a hot object will reach the ambient temperature after a finite amount of time.

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

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).

Textbook Question

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