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 9.2.36b

Chapter 9, Problem 9.2.36b

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).

y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)

Verified step by step guidance

Identify the given differential equation and initial condition: \(y'(t) = \frac{t}{y}\) with \(y(0) = 4\).

Note the exact solution provided: \(y(t) = \sqrt{t^2 + 16}\), which will be used to find the error at \(T = 2\).

Use Euler's method with step size \(\Delta t = 0.1\) to approximate \(y(2)\). Recall Euler's formula: \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\), where \(f(t, y) = \frac{t}{y}\).

Starting from \(t_0 = 0\) and \(y_0 = 4\), iteratively compute \(y_1, y_2, \ldots, y_{20}\) (since \(\frac{2}{0.1} = 20\) steps) using Euler's method.

Calculate the error at \(t = 2\) by subtracting the Euler approximation \(y_{20}\) from the exact solution \(y(2) = \sqrt{2^2 + 16}\): \(\text{Error} = |y(2) - y_{20}|\).

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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 first-order differential equations. It uses a step size Δt to iteratively estimate the value of the function at discrete points, starting from an initial condition. The method updates the solution by adding the product of the derivative and the step size to the previous value.

Exact Solution of Differential Equations

The exact solution is an explicit formula that satisfies the differential equation and initial conditions. In this problem, y(t) = √(t² + 16) is the exact solution, providing the true value of y at any time t. Comparing the exact solution to numerical approximations helps assess accuracy.

Error Analysis in Numerical Methods

Error analysis involves calculating the difference between the numerical approximation and the exact solution at a specific point, here at t = T. This error quantifies the accuracy of the approximation and helps in understanding how step size and method choice affect results.

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Related Practice

Textbook Question

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

b. The general solution of the separable equation y'(t) = t/(y' + 10y⁴) can be expressed explicitly with y in terms of t.

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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} Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is:

dM/dt = -rM(t)ln(M(t)/K), M(0) = M₀,

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

b. Graph the solution for M₀ = 100 and r = 0.05.

Textbook Question

Euler’s method on more general grids Suppose the solution of the initial value problem y'(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b].

b. Write the first step of Euler’s method to compute u1.

Textbook Question

{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation

m · v'(t) = mg + f(v)

mass | acceleration | external forces

where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.

c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a.

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Textbook Question

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.

c. Draw a representative direction field in the case that a<0. Show that if A>−b/a, then the solution decreases for t≥0, and that if A<−b/a, then the solution increases for t≥0.

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Textbook Question

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

c. The general solution of the equation yy'(x) = xe⁻ʸ can be found using integration by parts.