Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.2.49b

Chapter 9, Problem 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, ...

Verified step by step guidance

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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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 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.

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Verifying Solutions of Differential Equations

Related Practice

Textbook Question

{Use of Tech} Intravenous drug dosing The amount of drug in the blood of a patient (in milligrams) administered via an intravenous line is governed by the initial value problem y’(t) = -0.02y + 3, y(0) = 0 where t is measured in hours.

b. What is the steady-state level of the drug?

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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.

a. Find the equilibrium solutions.

y′(t) = y(y - 3)(y + 2)

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Textbook Question

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

b. If k>0 and b>0 then y(t)=0 is never a solution of y'(t)=ky−b.

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Textbook Question

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

a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.

y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

Textbook Question

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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Textbook Question

{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm.

b. Find the solution in k=0.1the case that and H=0.5m.

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