Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.2.32b

Chapter 9, Problem 9.2.32b

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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Identify the given differential equation and initial condition: $y'(t) = 2t + 1$ with $y(0) = 0$.

Note the exact solution provided: $y(t) = t^{2} + t$.

Recall Euler's method formula for approximating the solution at discrete points: $y_{n+1} = y_n + h \cdot f(t_n, y_n)$, where $h$ is the step size and $f(t, y) = y'(t)$.

Choose the step size $h$ based on the points of interest, here from $t=0$ to $t=0.2$ and then to $t=0.4$, so $h=0.2$. Use Euler's method iteratively to find approximate values $y(0.2)$ and $y(0.4)$.

Calculate the errors at $t=0.2$ and $t=0.4$ by subtracting the Euler approximations from the exact solution values: $\text{Error} = |y_{exact}(t) - y_{Euler}(t)|$.

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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 given by the differential equation at the current point. The accuracy depends on the step size, with smaller steps generally yielding better approximations.

Exact Solution and Error Calculation

The exact solution is the precise function satisfying the differential equation and initial condition. To compute errors in Euler's method, subtract the approximate value from the exact value at specific points. This difference quantifies the accuracy of the numerical approximation.

Initial Value Problems (IVPs)

An initial value problem specifies a differential equation along with a starting value for the unknown function. Solving an IVP involves finding a function that satisfies both the differential equation and the initial condition, providing a unique solution curve.

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Initial Value Problems

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.

{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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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

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, ...

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Textbook Question

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.

b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.

x′(t) = 2x − xy, y′(t) = −y + xy

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Textbook Question

brOrthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x² + y² = a²

b. The family of trajectories orthogonal to 2x² + y² = a² satisfies the differential equation dy/dx = y/(2x). Why?

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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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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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Key Concepts

Euler's Method

Exact Solution and Error Calculation

Initial Value Problems (IVPs)

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