Chapter 5

Introduction to Systems and Phase Plane Analysis

Educators

TS

Problem 1

$$\begin{array}{l}{y^{\prime \prime}(t)+t y^{\prime}(t)-3 y(t)=t^{2}} \\ {y(0)=3, \quad y^{\prime}(0)=-6}\end{array}$$

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Problem 2

$$\begin{array}{l}{y^{\prime \prime}(t)=\cos (t-y)+y^{2}(t)} \\ {y(0)=1, \quad y^{\prime}(0)=0}\end{array}
$$

Matt J.
Numerade Educator

Problem 3

$$\begin{array}{l}{y^{(4)}(t)-y^{(3)}(t)+7 y(t)=\cos t} \\ {y(0)=y^{\prime}(0)=1, \quad y^{\prime \prime}(0)=0, \quad y^{(3)}(0)=2}\end{array}$$

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Problem 4

$$\begin{array}{l}{y^{(6)}(t)=\left[y^{\prime}(t)\right]^{3}-\sin (y(t))+e^{2 t}} \\ {y(0)=y^{\prime}(0)=\cdots=y^{(5)}(0)=0}\end{array}$$

Matt J.
Numerade Educator

Problem 5

$$\begin{array}{ll}{x^{n}+y-x^{\prime}=2 t ;} & {x(3)=5, \quad x^{\prime}(3)=2} \\ {y^{\prime \prime}-x+y=-1 ;} & {y(3)=1,}\end{array}$$
$$\left[\text {Hint} : \operatorname{Set} x_{1}=x, \quad x_{2}=x^{\prime}, \quad x_{3}=y, \quad x_{4}=y^{\prime}\right]$$

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Problem 6

$$\begin{array}{ll}{3 x^{n}+5 x-2 y=0 ;} & {x(0)=-1, \quad x^{\prime}(0)=0} \\ {4 y^{\prime \prime}+2 y-6 x=0 ;} & {y(0)=1, \quad y^{\prime}(0)=2}\end{array}$$

Matt J.
Numerade Educator

Problem 7

$$\begin{array}{ll}{x^{m}-y=t ;} & {x(0)=x^{\prime}(0)=x^{\prime \prime}(0)=4} \\ {2 x^{\prime \prime}+5 y^{\prime \prime}-2 y=1 ;} & {y(0)=y^{\prime}(0)=1}\end{array}$$

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Problem 8

Sturm Liouville Form. A second-order equation is said to be in Sturm Liouville form if it is expressed as
$$\left[p(t) y^{\prime}(t)\right]^{\prime}+q(t) y(t)=0$$

Matt J.
Numerade Educator

Problem 8

SturmLiouville Form. A second-order equation is said to be in SturmLiouville form if it is expressed as
$$\left[p(t) y^{\prime}(t)\right]^{\prime}+q(t) y(t)=0$$
Show that the substitutions $x_{1}=y, x_{2}=p y^{\prime}$ result inthe normal form
$$\begin{aligned} x_{1}^{\prime} &=x_{2} / p \\ x_{2}^{\prime} &=-q x_{1} \end{aligned}$$

Matt J.
Numerade Educator

Problem 8

Sturm-Liouville Form. A second-order equation is said to be in Sturm-Liouville form if it is expressed as
$$\left[p(t) y^{\prime}(t)\right]^{\prime}+q(t) y(t)=0$$
Show that the substitutions $x_{1}=y, x_{2}=p y^{\prime}$ result in the normal form
$$\begin{aligned} x_{1}^{\prime} &=x_{2} / p \\ x_{2}^{\prime} &=-q x_{1} \end{aligned}$$
If $y(0)$ = $ a $ and $ y' $ 102 = $ b $ are the initial values for the Sturm-Liouville problem, what are
$x_{1}(0)$ and $x_{2}(0)?$

Matt J.
Numerade Educator

Problem 9

In Section 3.6, we discussed the improved Euler's method for approximating the solution to a first order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

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Problem 10

$$\begin{array}{l}{y^{\prime \prime}+t y^{\prime}+y=0} \\ {y(0)=1, \quad y^{\prime}(0)=0 \quad \text { on }[0,1]}\end{array}$$

Matt J.
Numerade Educator

Problem 11

$$\begin{array}{l}{\left(1+t^{2}\right) y^{\prime \prime}+y^{\prime}-y=0} \\ {y(0)=1, \quad y^{\prime}(0)=-1 \quad \text { on }[0,1]}\end{array}$$

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Problem 12

$$\begin{array}{l}{t^{2} y^{\prime \prime}+y=t+2} \\ {y(1)=1, \quad y^{\prime}(1)=-1 \quad \text { on }[1,2]}\end{array}$$

Matt J.
Numerade Educator

Problem 13

$$\begin{array}{l}{y^{\prime \prime}=t^{2}-y^{2}} \\ {y(0)=0, \quad y^{\prime}(0)=1 \quad \text { on }[0,1]}\end{array}$$
(Can you guess the solution?)

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Problem 14

Using the vectorized RungeKutta algorithm with $h=0.5$, approximate the solution to the initial value problem
$$\begin{array}{l}{3 t^{2} y^{n}-5 t y^{\prime}+5 y=0} \\ {y(1)=0, \quad y^{\prime}(1)=\frac{2}{3}}\end{array}$$
at $t=8$. Compare this approximation to the actual solution $y(t)=t^{3 / 3}-t$

Matt J.
Numerade Educator

Problem 15

Using the vectorized Runge-Kutta algorithm, approximate the solution to the initial value problem
$$y^{n}=t^{2}+y^{2} ; \quad y(0)=1, \quad y^{\prime}(0)=0$$
at $t=1$ 1. Starting with $h=1$, continue halving the step size until two successive approximations
[of both $y(1)$ and $y^{\prime}(1)$] differ by at most 0.01.

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Problem 16

Using the vectorized Runge-Kutta algorithm for systems with $h=0.125$ approximate the solution to the initial value problem
$$\begin{array}{ll}{x^{\prime}=2 x-y ;} & {x(0)=0} \\ {y^{\prime}=3 x+6 y ;} & {y(0)=-2}\end{array}$$
at $t=1$ Compare this approximation to the actual solution
$$x(t)=e^{5 t}-e^{3 t}, \quad y(t)=e^{3 t}-3 e^{5 t}$$

Matt J.
Numerade Educator

Problem 17

Using the vectorized Runge-Kutta algorithm, approximate the solution to the initial value problem
$$\begin{array}{ll}{\frac{d u}{d x}=3 u-4 v ;} & {u(0)=1} \\ {\frac{d v}{d x}=2 u-3 v ;} & {v(0)=1}\end{array}$$
at $x=1$. Starting with $h=1$ continue halving the step size until two successive approximations of $u(1)$ and $v(1)$ differ by at most 0.001.

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Problem 18

Combat Model. A simplified mathematical model for conventional versus guerrilla combat is given by the system
$$\begin{array}{ll}{x_{1}^{\prime}=-(0.1) x_{1} x_{2} ;} & {x_{1}(0)=10} \\ {x_{2}^{\prime}=-x_{1} ;} & {x_{2}(0)=15}\end{array}$$
where $x_{1}$ and $x_{2}$ are the strengths of guerrilla and conventional troops, respectively, and 0.1 and 1 are the $combat$ $effectiveness$ $coefficients$ Who will win the conflict: the conventional troops or the guerrillas? [Hint: Use the vectorized Runge-Kutta algorithm for systems with $h=0.1$ to approximate the solutions.]

Matt J.
Numerade Educator

Problem 19

Predator-Prey Model. The Volterra-Lotka predatorprey model predicts some rather interesting behavior that is evident in certain biological systems. For example, suppose you fix the initial population of prey but increase the initial population of predators. Then the population cycle for the
prey becomes more severe in the sense that there is a long period of time with a reduced population of prey followed by a short period when the population of prey is very large.To demonstrate this behavior, use the vectorized Runge-Kutta algorithm for systems with $h=0.5$ 0.5 to approximate
the populations of prey $x$ and of predators $y$ over the period 30, 54 that satisfy the Volterra-Lotka system
$$\begin{aligned} x^{\prime} &=x(3-y) \\ y^{\prime} &=y(x-3) \end{aligned}$$
under each of the following initial conditions:
$$\begin{array}{ll}{\text { (a) } x(0)=2,} & {y(0)=4} \\ {\text { (b) } x(0)=2,} & {y(0)=5} \\ {\text { (c) } x(0)=2,} & {y(0)=7}\end{array}$$

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Problem 20

In Project C of Chapter 4, it was shown that the simple pendulum equation
$$\theta^{\prime \prime}(t)+\sin \theta(t)=0$$
has periodic solutions when the initial displacement and velocity are small. Show that the period of the solution may depend on the initial conditions by using the vectorized Runge-Kutta algorithm with $h=0.02$ to approximate the solutions to the simple pendulum problem on [0, 4] for the initial conditions:
$$\begin{array}{l}{\text { (a) } \theta(0)=0.1, \quad \theta^{\prime}(0)=0} \\ {\text { (b) } \theta(0)=0.5, \quad \theta^{\prime}(0)=0} \\ {\text { (c) } \theta(0)=1.0,} \\ {[\text { Hint: Approximate the length of time it takes to reach }} \\ {-\theta(0) .}\end{array}$$

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Problem 21

Fluid Ejection. In the design of a sewage treatment plant, the following equation arises:
$$\begin{array}{l}{60-H=(77.7) H^{\prime \prime}+(19.42)\left(H^{\prime}\right)^{2}} \\ {H(0)=H^{\prime}(0)=0}\end{array}$$
where $H$ is the level of the fluid in an ejection chamber and $t$ is the time in seconds. Use the vectorized Runge-Kutta algorithm with $h=0.5$ to approximate $H(t)$ over the interval $[0,5]$ .

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Problem 22

Oscillations and Nonlinear Equations. For the initial
value problem
$$\begin{array}{l}{x^{\prime \prime \prime}+(0.1)\left(1-x^{2}\right) x^{\prime}+x=0} \\ {x(0)=x_{0}, \quad x^{\prime}(0)=0}\end{array}$$
use the vectorized Runge-Kutta algorithm with $h=0.02$
to illustrate that as $t$ increases from 0 to 20, the solution
$x$ exhibits damped oscillations when $x_{0}=1,$ whereas $x$
exhibits expanding oscillations when $x_{0}=2.1$

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Problem 23

Nonlinear Spring. The Duffing equation
$$
y^{\prime \prime}+y+r y^{3}=0
$$
where $r$ is a constant, is a model for the vibrations of a
mass attached to a nonlinear spring. For this model, does
the period of vibration vary as the parameter $r$ is varied?
Does the period vary as the initial conditions are varied?
[Hint: Use the vectorized Runge-Kutta algorithm with
$h=0.1$ to approximate the solutions for $r=1$ and 2
with initial conditions $y(0)=a, y^{\prime}(0)=0$ for $a=1$
$2,$ and 3.1

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Problem 24

Pendulum with Varying Length. A pendulum is formed by a mass m attached to the end of a wire that is attached to the ceiling. Assume that the length $I(t)$ of the wire varies with time in some predetermined fashion. If $\theta(t)$ is the angle in radians between the pendulum and the vertical, then the motion of the pendulum is governed for small angles by the initial value problem
$$\begin{array}{l}{l^{2}(t) \theta^{\prime \prime}(t)+2 l(t) l^{\prime}(t) \theta^{\prime}(t)+g l(t) \sin (\theta(t))=0} \\ {\theta(0)=\theta_{0}, \quad \theta^{\prime}(0)=\theta_{1}}\end{array}$$
where g is the acceleration due to gravity. Assume that
$$l(t)=I_{0}+l_{1} \cos (\omega t-\phi)$$
where $l_{1}$ is much smaller than $l_{0}$ l0. (This might be a model
for a person on a swing, where the $pumping$ action
changes the distance from the center of mass of the swing to the point where the swing is attached.) To simplify the computations, take $g=1$Using the Runge-Kutta algorithm with $h=0.1$ study the motion of the pendulum when $\theta_{0}=0.05, \theta_{1}=0, \quad l_{0}=1, l_{1}=0.1$ $\omega=1,$ and $\phi=0.02 .$ In particular, does the pendulum
ever attain an angle greater in absolute value than the
initial angle $\theta_{0} ?$

TS
Tim S.
Numerade Educator

Problem 25

Using the Runge-Kutta algorithm for systems with $h=0.05,$ approximate the solution to the initial value problem
$$\begin{array}{l}{y^{\prime \prime \prime}+y^{\prime \prime}+y^{2}=t} \\ {y(0)=1, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(0)=1}\end{array}$$
at $t=1$.

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Problem 26

Use the Runge-Kutta algorithm for systems with $h=0.1$
to approximate the solution to the initial value problem
$$
\begin{array}{ll}{x^{\prime}=y z ;} & {x(0)=0} \\ {y^{\prime}=-x z ;} & {y(0)=1} \\ {z^{\prime}=-x y / 2 ;} & {z(0)=1}\end{array}
$$
at $t=1$.

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Problem 27

Generalized Blasius Equation. H. Blasius, in his study of
laminar flow of a fluid, encountered an equation of the form
$$
y^{\prime \prime \prime}+y y^{\prime \prime}=\left(y^{\prime}\right)^{2}-1
$$
Use the Runge-Kutta algorithm for systems with $h=0.1$
to approximate the solution that satisfies the initial con-
ditions $y(0)=0, y^{\prime}(0)=0,$ and $y^{\prime \prime}(0)=1.32824$
Sketch this solution on the interval $[0,2]$ .

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Problem 28

Lunar Orbit. The motion of a moon moving in a planar
orbit about a planet is governed by the equations
$$
\frac{d^{2} x}{d t^{2}}=-G \frac{m x}{r^{3}}, \quad \frac{d^{2} y}{d t^{2}}=-G \frac{m y}{r^{3}}
$$
where $r :=\left(x^{2}+y^{2}\right)^{1 / 2}, G$ is the gravitational constant,
and $m$ is the mass of the planet. Assume $G m=1 .$ When
$x(0)=1, x^{\prime}(0)=y(0)=0,$ and $y^{\prime}(0)=1,$ the
motion is a circular orbit of radius 1 and period 2$\pi$ .
(a) Setting $x_{1}=x, x_{2}=x^{\prime}, x_{3}=y, x_{4}=y^{\prime},$ express
the governing equations as a first-order system in
normal form.
(b) Using $h=2 \pi / 100 \approx 0.0628318$ , compute one
orbit of this moon (i.e., do $N=100$ steps.). Do your
approximations agree with the fact that the orbit is a
circle of radius 1?

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Problem 29

Competing Species. Let $p_{i}(t)$ denote, respectively, the
populations of three competing species $S_{i, i}=1,2,3$ .Suppose these species have the same growth rates, and
the maximum population that the habitat can support is
the same for each species. (We assume it to be one unit.)
Also suppose the competitive advantage that $S_{1}$ has over
$S_{2}$ is the same as that of $S_{2}$ over $S_{3}$ and $S_{3}$ over $S_{1}$ . This
situation is modeled by the system
$$
\begin{aligned} p_{1}^{\prime} &=p_{1}\left(1-p_{1}-a p_{2}-b p_{3}\right) \\ p_{2}^{\prime} &=p_{2}\left(1-b p_{1}-p_{2}-a p_{3}\right) \\ p_{3}^{\prime} &=p_{3}\left(1-a p_{1}-b p_{2}-p_{3}\right) \end{aligned}
$$where $a$ and $b$ are positive constants. To demonstrate the
population dynamics of this system when $a=b=0.5$
use the Runge-Kutta algorithm for systems with $h=0.1$
to approximate the populations $p_{i}$ over the time interval
$[0,10]$ under each of the following initial conditions:
$$
\begin{array}{ll}{\text { (a) } p_{1}(0)=1.0,} & {p_{2}(0)=0.1, \quad p_{3}(0)=0.1} \\ {\text { (b) } p_{1}(0)=0.1,} & {p_{2}(0)=1.0,} & {p_{3}(0)=0.1} \\ {\text { (c) } p_{1}(0)=0.1,} & {p_{2}(0)=0.1, \quad p_{3}(0)=1.0}\end{array}$$
On the basis of the results of parts $(a)-(c),$ decide what
you think will happen to these populations as $t \rightarrow+\infty$

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Problem 30

Spring Pendulum. Let a mass be attached to one end of
a spring with spring constant $k$ and the other end attached
to the ceiling. Let $l_{0}$ be the natural length of the spring
and let $l(t)$ be its length at time $t .$ If $\theta(t)$ is the anglebetween the pendulum and the vertical, then the motion of the spring pendulum is governed by the system
$$
\begin{array}{l}{l^{\prime \prime}(t)-l(t) \theta^{\prime}(t)-g \cos \theta(t)+\frac{k}{m}\left(l-l_{0}\right)=0} \\ {l^{2}(t) \theta^{\prime \prime}(t)+2 l(t) I^{\prime}(t) \theta^{\prime}(t)+g l(t) \sin \theta(t)=0}\end{array}
$$
Assume $g=1, k=m=1,$ and $l_{0}=4 .$ When the sys-
tem is at rest, $l=l_{0}+m g / k=5 .$
(a) Describe the motion of the pendulum when
$l(0)=5.5, l^{\prime}(0)=0, \theta(0)=0,$ and $\theta^{\prime}(0)=0$
(b) When the pendulum is both stretched and given an
angular displacement, the motion of the pendulum is
more complicated. Using the Runge-Kutta algorithm for systems with $h=0.1$ to approximate the solution, sketch the graphs of the length $l$ and the angular
displacement $\theta$ on the interval $[0,10]$ if $l(0)=5.5$
$l^{\prime}(0)=0, \theta(0)=0.5,$ and $\theta^{\prime}(0)=0$

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