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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 9.2.32b
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

Verified step by step guidance
1
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.
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Euler's Method

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.
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Determining Error and Relative Error

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

Properties of stirred tank solutions


b. Verify that M(0) = M₀

Textbook Question

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

b. Euler’s method is used to compute exact values of the solution of an initial value problem. 

Textbook Question

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


b. The solution of a stirred tank initial value problem always approaches a constant as t→∞

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Textbook Question

{Use of Tech} Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t≥0. The relevant initial value problem is 

dM/dt=−rM ln(M/K),M(0)=M0, 

where r and K are positive constants and 0<M0<K.

b. Solve the initial value problem and graph the solution for r=1,K=4, and M0=1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor? 

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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