DE-BOOK Chapter 13 Exercises

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Section Chapter 13 Exercises

Reading Questions ᯓ★❓ Quick-Answer Questions

1. Multiple-Choice.

2. Short-Answer.

Exercises Exercises

Euler’s Method for Systems.

Consider the initial-valued system

\begin{align*} y' \amp = 3z,\quad \amp y(2) \amp = 3, \\ z' \amp = y+z^2, \amp z(2) \amp = -4. \end{align*}

and answer the following:

1.

Compute two iterations of Euler’s method using step size \(\ds h = 0.1. \)

2.

What is the meaning of your answer?

3. Euler’s Method for Systems.

Consider the initial-valued system

\begin{align*} x' \amp = 2x - y + t, \amp x(0) = 6, \\ y' \amp = x, \amp y(0) = 2. \end{align*}

Approximate \(\ds x(0.2) \) and \(\ds y(0.2) \) using \(\Delta t = 0.1\text{.}\)

Reduction of Order.

Reduce each of the following to a system of first-order differential equations.

4.

\(\ds y'' - 4y' + 4y = 0 \)

5.

\(\ds x^2 y''' = 2y' \)

6.

\(\ds y'' + x = 12 \ds x' + 2yt = \cos t \)

Basic Euler’s Method
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🔗

Consider the IVP below.\(\dsy' = 2x - 3y + 1,\quad y(1) = 5\)

7.

Euler’s Method for Higher Order DEs \amp Systems Consider the IVP below. \(\ds y'' + 2y' + y^2 = 0\quad \ds y(0.2) \) using step size \(\ds h = 0.1 \) .

Basic Euler’s Method
Use Euler’s Method to approximate \(\ds y(3.2) \) wher \(\ds y \) is the solution to \(\ds y'' - 2y = e^{t-3} \cos t,\quad y(3) = -1,\quad y'(3) = 0, \)
🔗

🔗

Consider the IVP below.\(\dsy' = 2x - 3y + 1,\quad y(1) = 5\)

8.

usin \(\ds \Delta t = 0.2. \) \label{partA_euler}

9.

using \(\ds \Delta t = 0.1 \)

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