Computer Algebra Recipes

Richard H Enns, George C McGuire

Chapter 3 Linear ODE Models - all with Video Answers

Educators

Chapter Questions

Problem 1

Modify the forcing function in the text recipe to include four heartbeats. Do an Internet search on blood pressure and discuss how the model could be improved.

Arun Bana

Numerade Educator

01:25

A simple electrical circuit consists of a resistor of $R$ ohms $(\Omega)$ in series with an inductor of $L$ henries $(\mathrm{H})$ and a battery providing a constant voltage $v$ (in volts $(\mathrm{V}))$. If $i(t)$ is the current in the circuit at time $t$, the voltage drop across the resistor is $R i$ (Ohm's law) and across the inductor is $L(d i / d t)$.
(a) Making use of Kirchhoff's voltage rule, which states that the sum of the voltage drops is equal to the supplied voltage, derive the first-order ODE for $i(t)$ and solve it, given that the initial current is zero.
(b) Taking $L=4 \mathrm{H}, R=12 \Omega$, and $v=60 \mathrm{~V}$, plot the solution over a time interval that brings the current to within $1 \%$ of its asymptotic value.
(c) The battery is replaced with a generator that produces a variable voltage of $v(t)=60 \sin (30 t)$ volts, the resistor and inductor retaining the same values as in part (b). If the initial current is zero, determine the current $i(t)$ for $t \geq 0$. Plot the current over a time interval that brings the current to within $1 \%$ of steady state.

Vishal Gupta

Numerade Educator

05:02

Problem 3

Learning curves are used by scientists interested in learning theory. A learning curve is a plot of the performance $P(t)$ of someone learning a skill as a function of the training time $t$. A simple model of learning curve is to assume that the rate $d P / d t$ at which performance improves is proportional to $M-P(t)$, where $M$ is the maximum level of performance. Determine the analytic form of $P(t)$ and discuss what this model implies.
Two new workers (Jimbo and Jumbo) are hired for an assembly line producing widgets. Jimbo (Jumbo) produces 25 (35) widgets during the first hour and 45 (50) widgets the second hour. Estimate the maximum number of widgets per hour that each is capable of producing and plot their learning curves.

Kevin Lu

Numerade Educator

04:23

Problem 4

A deer population initially numbers 1000 and has a growth rate of 0.5 when $t$ is measured in months. The population is "harvested" throughout the year at the rate of $h(2+\cos (\pi t / 6))$ per month, where $h$ is a constant. Determine the deer population number as a function of $t$. What value of $h$ would lead to a zero population growth over the 12 -month period from $t=0$ to $t=12$ ? Plot the deer-population curve for this 12 -month period.

Himanshu Kushwaha

Numerade Educator

06:44

Problem 5

Solve Professor Nerd's problem directly with the dsolve command and check that the answer is identical to that derived in the text recipe.

Stanley Enemuo

Numerade Educator

01:34

Problem 6

The drive force in the text recipe is replaced with $F_{\text {drive }}=\sin (t) t^2 e^{-t}$. Determine the velocity as a function of time and plot the result over the interval $t=0$ to 10 . What is Evil Knievel's maximum velocity and at what time does this occur? What is his maximum displacement from the origin in this interval and at what time does this occur?

R M

Numerade Educator

09:46

Problem 7

With all other parameters the same as in the text recipe, what is the maximum value of the drag coefficient $k$ such that the ball just clears the fence? What are the two corresponding initial angles?

Zulfiqar Ali

Numerade Educator

09:46

Problem 8

If the drag coefficient $k$ is equal to $0.1 \mathrm{~s}^{-1}$, how high can the fence be for the ball to just clear it, all other parameters the same as in the text recipe? What are the two initial angles in this case?

Zulfiqar Ali

Numerade Educator

07:43

Problem 9

For a sphere of diameter $d$ meters moving in air, the approximate value of the constant $k$ in Stokes's linear resistance law, $\vec{F}_{\text {drag }}=-k \vec{v}$ newtons, is given by $k=1.55 \times 10^{-4} d$. For a small spherical raindrop (density $\rho=10^3 \mathrm{~kg} / \mathrm{m}^3$, $d=10^{-4} \mathrm{~m}$ ) falling from rest, determine the distance through which it falls in $t$ seconds and its velocity then. Plot the distance and velocity separately over a time interval $t=0$ to the time at which the velocity reaches 99 percent of its terminal (asymptotic) value.

Yogesh Sawant

Numerade Educator

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Problem 10

Making use of the integral transform package, mimic a hand calculation to derive the solution for the second example.

Victor Salazar

Numerade Educator

08:53

Problem 11

Use the integral transform package and the Laplace transform method to solve the following ODEs. Confirm the solutions using the method= laplace option in the dsolve command. Plot each solution over a suitable time range that includes the steady-state regime. Extract the transient part of the solution and estimate the time interval over which it lasts.
(a) $\dot{y}+5 y=\cos (t)+e^{-t}, \quad y(0)=1$;
(b) $\ddot{y}-5 \dot{y}+6 y=0, \quad y(0)=2, \dot{y}(0)=5$;
(c) $9 \ddot{y}+6 \dot{y}+y=5, \quad y(0)=6, \dot{y}(0)=1$;
(d) $5 \ddot{y}+2 \dot{y}+y=\sin ^2(t), \quad y(0)=-3, \dot{y}(0)=1$;
(e) $\dot{x}=-4 x+y+3, \quad \dot{y}=-4 x-4 y+5, \quad x(0)=1, y(0)=-1$;
(f) $\ddot{y}+\dot{y}=2 \cos ^4(t) e^{-t}, \quad y(0)=0, \dot{y}(0)=-2$;
(g) $\ddot{y}+2 \dot{y}=t \cos (t) e^{-t}+t^2 \cos (2 t) e^{-2 t}, \quad y(0)=0, \dot{y}(0)=0$.

Nadir Musofer

Numerade Educator

01:32

Problem 12

Consider the following linear second-order ODE:
$$
x y^{\prime \prime}(x)+y^{\prime}(x)+x y(x)=0, \quad y(0)=1, y^{\prime}(0)=0 .
$$
(a) Show that Maple is able to produce a closed-form analytic solution. Identify the ODE and the "special" function that appears in the solution.
(b) Derive a series solution for $y(x)$, keeping sufficient terms to achieve $1 \%$ agreement with the analytic answer at $x=X=10$.
(c) Plot the series derived in part (b) and the analytic solution together in the same figure over the range $x=0$ to $X$, representing the analytic answer by appropriately sized and spaced circles.

Narayan Hari

Numerade Educator

33:31

Problem 13

Consider the following linear second-order ODE:
$$
x y^{\prime \prime}(x)+2 y^{\prime}(x)+x y(x)=0, \quad y(0)=1, y^{\prime}(0)=0 .
$$
(a) Show that Maple is able to produce a closed-form analytic solution.
(b) Derive a series solution for $y(x)$, keeping sufficient terms to achieve $1 \%$ agreement with the analytic answer at $x=X=20$.
(c) Plot the series derived in part (b) and the analytic solution together in the same figure over the range $x=0$ to $X$, representing the analytic answer by appropriately sized and spaced circles.

Chris Trentman

Numerade Educator

00:55

Problem 14

Carry out the same steps as in Jennifer's text problem for the ODE
$$
y^{\prime \prime}(x)-x y^{\prime}(x)-1=0, \quad y(0)=1, y^{\prime}(0)=0 .
$$

Keshav Singh

Numerade Educator

02:12

Problem 15

Show that the following ODE is of the Sturm-Liouville form:
$$
y^{\prime \prime}(x)-2 x y^{\prime}(x)+2 n y(x)=0 .
$$
Determine the general solution of this ODE and plot the included special functions over a suitable range of the independent variable $x$.

Manik Pulyani

Numerade Educator

08:06

Problem 16

Bessel functions of different orders can be related through recurrence relations. Use Maple to prove the following Bessel function recurrence relations.
$\bullet$ $J_{m-1}(x)+J_{m+1}(x)=(2 m / x) J_m(x)$;
$\bullet$ $4 \frac{d^2 J_m(x)}{d x^2}=J_{m+2}(x)-2 J_m(x)+J_{m-2}(x)$.

Wen Zheng

Numerade Educator

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Problem 17

Find the general solution of each of the following ODEs in terms of Bessel functions and identify the order. Identify any other new functions that occur.
(a) $y^{\prime \prime}(x)+y(x) / \sqrt{x}=0$;
(b) $y^{\prime \prime}(x)+x y(x)=0$, Hint: Use convert ( , Bessel);
(c) $\frac{d^2}{d x^2}\left(x^{\frac{16}{5}} \frac{d^2 y}{d x^2}\right)-x^{\frac{8}{5}} y(x)=0$;

Victor Salazar

Numerade Educator

03:14

Problem 18

An important general property that all solutions $y_n(x)$ of the Sturm-Liouville equation corresponding to a given $\lambda_n$ possess is orthogonality. Provided that $y(x)$ or $y^{\prime}(x)$ or $p(x)$ vanishes at the endpoints $a$ and $b$ of the range (referred to as Sturm-Liouville boundary conditions), then
$$
\int_a^b w(x) y_m(x) y_n(x) d x=0, \quad \text { for } m \neq n,
$$
where $w(x)$ is referred to as the weight function. Confirm the orthogonality property for Legendre functions of the second kind of orders 2 and 3 over the range $x=-1$ to +1 . Which of the possible Sturm-Liouville boundary conditions is satisfied?

James Kiss

Numerade Educator

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Problem 19

Determine the transverse normal modes of vibration of the bungee cord if a mass $M=60 \mathrm{~kg}$ hangs from the lower end. Animate one of the normal modes.

Lainey Roebuck

Numerade Educator

01:51

Problem 20

A vibrating spring is governed by the ODE
$$
\ddot{x}(t)+a \dot{x}(t)+k x(t)=0,
$$
where the spring coefficient is $k=b+c e^{d t}$, and initially $x(0)=A, \dot{x}(0)=0$. Determine the analytic form of $x(t)$. Taking $a=\sqrt{5}, b=\frac{1}{4}, c=1, d=1$, and $A=1$, plot $x(t)$ over a time interval for which the oscillations effectively vanish. Determine the threshold on $\alpha$ for critical damping.

A M

Numerade Educator

12:12

Problem 21

Consider a pendulum that consists of a point mass $m$ at the bottom end of a light supporting rod of length $L$ that is allowed to move in a vertical plane about a pivot point at its top end. Suppose that $L$ increases at a steady rate, i.e., $L=L_0+v t$, where $v>0$ is a constant speed and $t$ the time. Letting the rod make an angle $\theta(t)$ with the vertical and neglecting drag, use Newton's second law to derive the ODE for small $\theta$. Solve the ODE for $L_0=1$ meter, $g=9.8 \mathrm{~m} / \mathrm{s}^2, \theta(0)=\pi / 6$ radians, $\dot{\theta}(0)=0$ radians $/ \mathrm{s}$, and $v=0.5 \mathrm{~m} / \mathrm{s}$. Animate the motion of the pendulum arm (representing it as a thick line) over the time interval $t=0$ to 100 seconds, taking 100 frames and using constrained scaling.

Tim Strang

Numerade Educator

10:42

Problem 22

A thin, vertical steel wire of length $L$ and circular cross section of radius $a$ is clamped at its bottom and is free at its top. Let $\theta$ be the angular deflection of the wire from the vertical at a distance $y$ from the top. If $L$ is small, the wire is stable in the vertical position, i.e., $\theta=0$ for all values of $y$. As $L$ increases, there is a critical value $L_{c r}$ beyond which the wire is unstable and will bend from the vertical.
The relevant ODE for small angular displacements $\theta$ is
$$
\frac{d^2 \theta}{d y^2}=-c^2 y \theta, \quad \text { where } c=\frac{2}{a} \sqrt{\frac{\rho g}{Y}} .
$$
Here $\rho$ is the mass density, $g$ is the acceleration due to gravity, and $Y$ is Young's modulus.
(a) Determine the solution of the ODE, subject to the boundary conditions $\theta=0$ at $y=L$ (wire clamped at bottom) and $d \theta / d y=0$ at $y=0$ (wire is free at top).
(b) Show that the onset of bending occurs at $L_{c r} \approx(2.8 / c)^{2 / 3}$.
(c) Determine $L_{c r}$ for a steel $\left(Y=2.1 \times 10^{11} \mathrm{~N} / \mathrm{m}^2\right.$ and $\left.\rho=7800 \mathrm{~kg} / \mathrm{m}^3\right)$ wire of radius $1 \mathrm{~mm}$. Take $g=9.8 \mathrm{~m} / \mathrm{s}^2$.

Alan Gavel

Numerade Educator

01:21

Problem 23

In the text recipe, at what critical value of $\Gamma$ does the oscillatory motion cease and the unit mass approach the equilibrium position monotonically?

Zachary Warner

Numerade Educator

03:27

Problem 24

Explore the text recipe for other parameter values and discuss any interesting behavior.

Ryan Pollard

Numerade Educator

04:21

Problem 25

Confirm that the sum of the reflection and transmission coefficients is equal to 1 for all $E$. You may do this either graphically or analytically.

Aniket Bajaj

Numerade Educator

03:43

Problem 26

Determine the energy at which the reflection and transmission coefficients are equal by (i) using the mouse, (ii) using the fsolve command.

Prachita Kush

Numerade Educator

02:24

Problem 27

Determine the transmission and reflection coefficients as a function of energy $E$ if the barrier has the form $V(x)=V e^{-x / L}$ between $x=0$ and $L=1$, with $V=1$. Outside the barrier region, $V(x)=0$. Plot the reflection and transmission coefficients in the same figure and discuss how the results compare with those obtained in the text recipe.

Aniket Bajaj

Numerade Educator