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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 9 - Differential Equations문제 9.2.31c
9장, 문제 9.2.31c

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


c. Which time step results in the more accurate approximation? Explain your observations.


y′(t) = 4−y, y(0) = 3; y(t) = 4−e⁻ᵗ

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Identify the given initial value problem (IVP): \(y'(t) = 4 - y\), with initial condition \(y(0) = 3\), and the exact solution \(y(t) = 4 - e^{-t}\).
Recall that Euler's method approximates the solution by using the formula \(y_{n+1} = y_n + h f(t_n, y_n)\), where \(h\) is the time step size and \(f(t, y) = 4 - y\) in this case.
Understand that the accuracy of Euler's method depends on the size of the time step \(h\): smaller \(h\) generally leads to a more accurate approximation because it reduces the local truncation error at each step.
To compare the accuracy for different time steps, compute the approximate values of \(y\) at a fixed time using Euler's method with each time step, then compare these approximations to the exact solution \(y(t) = 4 - e^{-t}\) at the same time.
Observe that the time step with the smaller \(h\) will produce values closer to the exact solution, demonstrating that a smaller time step results in a more accurate approximation in Euler's method.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Euler's Method

Euler's method is a numerical technique to approximate solutions of initial value problems for differential equations. It uses a stepwise approach, estimating the next value by moving along the slope (derivative) at the current point. The accuracy depends on the size of the time step chosen.
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07:33
Euler's Method

Time Step Size and Accuracy

The time step size in Euler's method determines the intervals at which the solution is approximated. Smaller time steps generally yield more accurate results because they better capture the curve's behavior, while larger steps can introduce significant errors and reduce precision.
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Euler's Method

Exact Solution and Error Analysis

Comparing the numerical approximation to the exact solution allows for error analysis. The exact solution y(t) = 4 - e^{-t} provides a benchmark to measure how close Euler's method approximations are, helping to understand how step size affects the error magnitude.
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Determining Error and Relative Error
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교과서 질문

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.

c. Sketch the solution curve that corresponds to the initial condition y0=1. 


y′(t) = 2y + 4

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교과서 질문

{Use of Tech} Free fall Using th e background given in Exercise 47, assume the resistance is given by f(v)=−Rv, for t≥0, where R>0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil).


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

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교과서 질문

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.

c. Find the equilibrium points for the system.


x′(t) = −3x + xy, y′(t) = 2y − xy

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교과서 질문

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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교과서 질문

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).

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교과서 질문

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


d. Sketch the direction field and verify that it is consistent with parts (a)–(c).


y'(t) = (y−2)(y+1)

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