Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 9 - Differential Equations

문제 9.2.49b

9장, 문제 9.2.49b

Convergence of Euler's method Suppose Euler's method is applied to the initial value problem y′(t) = ay, y(0) = 1, which has the exact solution y(t) = eᵃᵗ. For this exercise, let h denote the time step (rather than Δt). The grid points are then given by tₖ = kh. We let uₖ be the Euler approximation to the exact solution y(tₖ), for k = 0, 1, 2, ...
b. Show by substitution that uₖ = (1 + ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...

검증된 단계별 안내

Recall the Euler's method update formula for the initial value problem $y'(t) = ay$ with step size $h$: \[u_{k+1} = u_k + h f(t_k, u_k)\] Since $f(t, y) = ay$, this becomes: \[u_{k+1} = u_k + h a u_k = u_k (1 + a h)\]

Given the initial condition $u_0 = y(0) = 1$, we want to verify that the proposed solution \[u_k = (1 + a h)^k\] satisfies the Euler update formula for all $k = 0, 1, 2, \ldots$

Substitute $u_k = (1 + a h)^k$ into the right-hand side of the Euler update formula: \[u_k (1 + a h) = (1 + a h)^k (1 + a h) = (1 + a h)^{k+1}\]

Compare this with the left-hand side $u_{k+1}$, which according to the proposed solution is: \[u_{k+1} = (1 + a h)^{k+1}\]

Since both sides are equal, this confirms by substitution that $u_k = (1 + a h)^k$ satisfies the Euler method recurrence relation for all $k$.

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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 for approximating solutions to initial value problems of differential equations. It uses a stepwise approach, updating the solution by moving along the slope given by the differential equation at each grid point. The formula is u_{k+1} = u_k + h f(t_k, u_k), where h is the step size.

Initial Value Problem (IVP) and Exact Solution

An initial value problem specifies a differential equation along with a starting value, such as y'(t) = ay with y(0) = 1. The exact solution to this IVP is y(t) = e^{at}, which provides a benchmark to compare numerical approximations like Euler's method.

Substitution to Verify Solutions

Substitution involves plugging a proposed solution into the given equation to verify its validity. Here, substituting u_k = (1 + ah)^k into the Euler update formula confirms it satisfies the recurrence relation, demonstrating that this expression correctly represents the numerical solution at each step.

관련 실천

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.

a. Find the equilibrium solutions.

y′(t) = y(y - 3)(y + 2)

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Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.

a. Explain why y=−b/a is an equilibrium solution and corresponds to a horizontal line in the direction field.

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교과서 질문

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.

{Use of Tech} 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.

b. Graph the solution for I = 10 mg/hr and k = 0.05 hr⁻¹.

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A bad loan Consider a loan repayment plan described by the initial value problem

B'(t)=0.03B−600,B(0)=40,000,

where the amount borrowed is B(0)=\$40,000, the monthly payments are \$600, and B(t) is the unpaid balance in the loan.

b. What is the most that you can borrow under the terms of this loan without going further into debt each month?

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{Use of Tech} Chemical rate equations Let y(t) be t he concentration of a substance in a chemical reaction (typical units are moles/liter). The change in the concentration, under appropriate conditions, is modeled by the equation dy/dt=-ky^n for t≥0, where k>0 is a rate constant and the positive integer n is the order of the reaction.

b. Solve the initial value problem for a second-order reaction (n=2) assuming y(0)=y0.

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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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