Skip to main content
Contents Index
Search Book
Search Results:
No results.
Read aloud Readability settings Prev Up Next Profile title here \(\newcommand\DLGray{\color{Gray}}
\newcommand\DLO{\color{BurntOrange}}
\newcommand\DLRa{\color{WildStrawberry}}
\newcommand\DLGa{\color{Green}}
\newcommand\DLGb{\color{PineGreen}}
\newcommand\DLBa{\color{RoyalBlue}}
\newcommand\DLBb{\color{Cerulean}}
\newcommand\ds{\displaystyle}
\newcommand\ddx{\frac{d}{dx}}
\newcommand\os{\overset}
\newcommand\us{\underset}
\newcommand\ob{\overbrace}
\newcommand\obt{\overbracket}
\newcommand\ub{\underbrace}
\newcommand\ubt{\underbracket}
\newcommand\ul{\underline}
\newcommand\laplacesym{\mathscr{L}}
\newcommand\lap[1]{\laplacesym\left\{#1\right\}}
\newcommand\ilap[1]{\laplacesym^{-1}\left\{#1\right\}}
\newcommand\tikznode[3][]
{\tikz[remember picture,baseline=(#2.base)]
\node[minimum size=0pt,inner sep=0pt,#1](#2){#3};
}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\newcommand{\sfrac}[2]{{#1}/{#2}}
\)
Exercises 8.6 Exercises
📝: Note on Variables.
In these exercises, you’ll see both
\(t\) and
\(x\) used as the independent variable in differential equations. This is common in mathematics—the choice of variable name doesn’t change the method! Whether you see
\(y' = f(t, y)\) or
\(y' = f(x, y)\text{,}\) Euler’s method works exactly the same way. The important thing is to consistently use the variable indicated in each problem.
✍🏻 Problems.
Work through the following problems.
1.
Approximate the solution to the differential equation
\begin{equation*}
y' = y^2 - x, \quad y(0) = -1
\end{equation*}
over the interval \([0, 1]\) with a step size of \(h = 0.5\text{.}\)
Hint .
Start with the given initial condition
\(y_0 = -1\) at
\(x_0 = 0\text{.}\) Use Euler’s update formula:
\(y_{k+1} = y_k + h \cdot f(x_k, y_k)\text{,}\) where
\(f(x, y) = y^2 - x\text{.}\) You’ll need to compute two steps to reach
\(x = 1\text{.}\)
Answer .
Using Euler’s method with
\(h=0.5\text{:}\) \(y(0)=-1\text{,}\) \(y(0.5) \approx -0.5\text{,}\) \(y(1) \approx -0.625\text{.}\)
2.
Use Euler’s Method to approximate the solution to the initial-value problem,
\begin{align}
y'(x) - 2x y(x) = 0 \amp \tag{8.5}\\
y(0) = 2, \amp \tag{8.6}
\end{align}
at the \(x\) -values \(0,\ 0.5,\ 1,\ 1.5,\ 2\) (spaced 0.5 apart).
Answer .
Approximate values:
\(y(0)=2\text{,}\) \(y(0.5)=2\text{,}\) \(y(1)=3\text{,}\) \(y(1.5)=6\text{,}\) \(y(2)=15\text{.}\)
3. Basic Euler’s Method.
Consider the initial-value problem
\begin{equation*}
y' = 2x - 3y + 1,\quad y(1) = 5
\end{equation*}
and answer the following:
(a) Compute 2 iterations of Euler’s method using step size
\(h = 0.1\text{.}\)
Answer .
\(\ds x_1 = 1.1 \hspace{0.5cm} \ds x_2 = 1.2, \hspace{0.5cm} \ds y_1 = 3.8, \hspace{0.5cm} \ds y_2 = 2.98 \)
(b) What is the meaning of your answer in part (a)?
(c) Compute 2 iterations of Euler’s method using step size
\(\ds h = 0.05\text{.}\)
Answer .
\(x_1 = 1.05 \hspace{0.5cm} x_2 = 1.1, \hspace{0.5cm} y_1 = 4.4, \hspace{0.5cm} y_2 = 3.895 \)
(d) What is the meaning of your answer in part (c)?
(e) Find the analytic solution to the IVP. Use your solution to compare the exact value of
\(y(1.1)\) with your answers from parts (a) and (c).
Answer .
\(\ds y = \frac{2}{3}x + \frac{1}{9} + \frac{38e^3}{9}e^{-3x},\quad y(1.1) \approx 3.9723 \)
4. Basic Euler’s Method.
Consider the initial-value problem
\begin{equation*}
y' = y,\quad y(0) = 1
\end{equation*}
and answer the following:
(a) Use Euler’s method to approximate
\(y(0.5)\) using step size
\(\ds h = 0.5\text{.}\)
(b) Use Euler’s method to approximate
\(y(0.5)\) using step size
\(h = 0.25\text{.}\)
Answer .
\(y(0.5) \approx 1.5625 \)
(c) Which of the approximations above do you trust more?
Answer .
The
\(h = 0.25\) approximation — smaller steps accumulate less error.
(d) Find the analytic solution to the IVP. Use it to compute the exact value of
\(y(0.5)\) and compare with your answers from parts (a) and (b).
Answer .
\(y = e^x,\quad y(0.5) \approx e^{0.5} \approx 1.6487\)
5. Comparing Step Sizes.
Consider the initial-value problem
\begin{equation*}
y' = -y, \quad y(0) = 1
\end{equation*}
on the interval \([0, 1]\text{.}\)
(a) Use Euler’s method with step size
\(h = 0.5\) to approximate
\(y(1)\text{.}\)
(b) Use Euler’s method with step size
\(h = 0.25\) to approximate
\(y(1)\text{.}\)
(c) The exact solution is
\(y(t) = e^{-t}\text{.}\) Calculate
\(y(1)\) exactly and compare with your approximations from parts (a) and (b). Which approximation is closer to the true value?
Answer .
\(y(1) = e^{-1} \approx 0.368\text{;}\) the smaller step size
\(h = 0.25\) gives a more accurate approximation.
6. Conceptual Understanding.
Answer the following questions about Euler’s method:
Why does decreasing the step size \(h\) generally improve the accuracy of Euler’s method? Explain in terms of the geometry of the solution curve.
Suppose you’re using Euler’s method and notice that your approximation seems to drift further from the true solution as you take more steps. What might be happening, and what could you do to improve the approximation?
Can Euler’s method produce the exact solution for any differential equation? If so, give an example. If not, explain why not.
Answer .
(a) Smaller steps keep the straight-line segments closer to the curved solution, so less error accumulates. (b) Errors compound step by step; use a smaller
\(h\) or a more accurate method. (c) Only when the true solution is a straight line (e.g.,
\(y' = c\) ) — otherwise every step introduces some error.
You have attempted
of
activities on this page.
