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Elementary Differential Equations with Boundary Value Problems

Werner E. Kohler, Lee W. Johnson

Chapter 6

Nonlinear Systems - all with Video Answers

Educators

Section 1

Introduction

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Problem 1

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
y^{\prime \prime}+t y^{\prime}+2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=2
$$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 2

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
y^{\prime \prime}+e^{t} y=\ln |t|, \quad y(-1)=0, \quad y^{\prime}(-1)=-1
$$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 3

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
y^{\prime \prime}+t y=\sin y^{\prime}, \quad y(0)=0, \quad y^{\prime}(0)=1
$$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 4

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
y^{\prime \prime}+\left(y^{\prime}\right)^{3}+y^{1 / 3}=\tan (t / 2), \quad y(1)=1, \quad y^{\prime}(1)=-2
$$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 5

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
t y^{\prime \prime}+\frac{1}{1+y+2 y^{\prime}}=e^{-t}, \quad y(2)=2, \quad y^{\prime}(2)=1
$$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 6

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
y^{\prime \prime \prime}+t^{2} y^{\prime \prime}=\sin t, \quad y(1)=0, \quad y^{\prime}(1)=1, \quad y^{\prime \prime}(1)=-1
$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 7

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
y^{\prime \prime \prime}+y^{\prime}+y^{2}=0, \quad y(-1)=0, \quad y^{\prime}(-1)=1, \quad y^{\prime \prime}(-1)=0
$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 8

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
y^{\prime \prime \prime}+\cos \left(t y^{\prime}\right)=t\left(y^{\prime \prime}\right)^{2}, \quad y(0)=1, \quad y^{\prime}(0)=1, \quad y^{\prime \prime}(0)=-2
$$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 9

In each exercise,
(a) Rewrite the given $n$th order scalar initial value problem as $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}$, by defining $y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)$ and
defining $y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)$ $\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\ y_{n}(t)\end{array}\right]$
(b) Compute the $n^{2}$ partial derivatives $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n$.
(c) For the system obtained in part (a), determine where in $(n+1)$-dimensional $t \mathbf{y}$-space the hypotheses of Theorem $6.1$ are not satisfied. In other words, at what points $\left(t, y_{1}, \ldots, y_{n}\right)$, if any, does at least one component function $f_{i}\left(t, y_{1}, \ldots, y_{n}\right)$ and/or at least one partial derivative function $\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n$ fail to be continuous? What is the largest open rectangular region $R$ where the hypotheses of Theorem $6.1$ hold?
$$
y^{\prime \prime \prime}+\frac{2 t^{1 / 3}}{(y-2)\left(y^{\prime \prime}+2\right)}=0, \quad y(0)=0, \quad y^{\prime}(0)=2, \quad y^{\prime \prime}(0)=2
$$

Victor Salazar
Victor Salazar
Numerade Educator
06:00

Problem 10

In each exercise, an initial value problem for a first order nonlinear system is given. Rewrite the problem as an equivalent initial value problem for a higher order nonlinear scalar differential equation.
$$
\frac{d}{d t}\left[\begin{array}{l}
y_{1} \\
y_{2}
\end{array}\right]=\left[\begin{array}{c}
y_{2} \\
t \cos ^{2}\left(y_{2}\right)-3 y_{1}+t^{4}
\end{array}\right], \quad\left[\begin{array}{l}
y_{1}(2) \\
y_{2}(2)
\end{array}\right]=\left[\begin{array}{r}
1 \\
-1
\end{array}\right]
$$

Uma Kumari
Uma Kumari
Numerade Educator
06:32

Problem 11

In each exercise, an initial value problem for a first order nonlinear system is given. Rewrite the problem as an equivalent initial value problem for a higher order nonlinear scalar differential equation.
$$
\frac{d}{d t}\left[\begin{array}{l}
y_{1} \\
y_{2}
\end{array}\right]=\left[\begin{array}{c}
y_{2} \\
y_{2} \tan \left(y_{1}\right)+e^{y_{2}}
\end{array}\right], \quad\left[\begin{array}{l}
y_{1}(0) \\
y_{2}(0)
\end{array}\right]=\left[\begin{array}{l}
0 \\
1
\end{array}\right]
$$

Uma Kumari
Uma Kumari
Numerade Educator
06:31

Problem 12

In each exercise, an initial value problem for a first order nonlinear system is given. Rewrite the problem as an equivalent initial value problem for a higher order nonlinear scalar differential equation.
$$
\frac{d}{d t}\left[\begin{array}{l}
y_{1} \\
y_{2} \\
y_{3}
\end{array}\right]=\left[\begin{array}{c}
y_{2} \\
y_{3} \\
y_{1} y_{2}+y_{3}^{2}
\end{array}\right], \quad\left[\begin{array}{l}
y_{1}(-1) \\
y_{2}(-1) \\
y_{3}(-1)
\end{array}\right]=\left[\begin{array}{r}
-1 \\
2 \\
-4
\end{array}\right]
$$

Uma Kumari
Uma Kumari
Numerade Educator
02:04

Problem 13

In each exercise, an initial value problem for a first order nonlinear system is given. Rewrite the problem as an equivalent initial value problem for a higher order nonlinear scalar differential equation.
$$
\frac{d}{d t}\left[\begin{array}{l}
y_{1} \\
y_{2} \\
y_{3}
\end{array}\right]=\left[\begin{array}{c}
y_{2} \\
y_{3} \\
\sqrt{y_{2} y_{3}+t^{2}}
\end{array}\right], \quad\left[\begin{array}{l}
y_{1}(1) \\
y_{2}(1) \\
y_{3}(1)
\end{array}\right]=\left[\begin{array}{l}
1 \\
\frac{1}{2} \\
3
\end{array}\right]
$$

Uma Kumari
Uma Kumari
Numerade Educator
10:27

Problem 14

Consider the initial value problem
$$
\frac{d}{d t}\left[\begin{array}{l}
y_{1} \\
y_{2}
\end{array}\right]=\left[\begin{array}{c}
\frac{5}{4} y_{1}^{1 / 5}+y_{2}^{2} \\
3 y_{1} y_{2}
\end{array}\right], \quad\left[\begin{array}{l}
y_{1}(0) \\
y_{2}(0)
\end{array}\right]=\left[\begin{array}{l}
0 \\
0
\end{array}\right]
$$
For the given autonomous system, the two functions $f_{1}\left(y_{1}, y_{2}\right)=\frac{5}{4} y_{1}^{1 / 5}+y_{2}^{2}$ and $f_{2}\left(y_{1}, y_{2}\right)=3 y_{1} y_{2}$ are continuous functions for all $\left(y_{1}, y_{2}\right)$.
(a) Show by direct substitution that
$$
y_{1}(t)=\left\{\begin{array}{lr}
0, & -\infty<t \leq c, \\
(t-c)^{5 / 4}, & c<t<\infty,
\end{array} \quad y_{2}(t)=0\right.
$$
is a solution of this initial value problem on $-\infty<t<\infty$ for any positive constant $c$.
(b) Since $c$ is an arbitrary positive constant, the solution of the given initial value problem is clearly not unique. Does this example contradict Theorem 6.1? Explain your answer.

Payton Sawyer
Payton Sawyer
Numerade Educator
10:28

Problem 15

Consider the initial value problem $y^{\prime \prime}+y^{2}=t, y(0)=y_{0}, y^{\prime}(0)=y_{0}^{\prime} .$ Can Laplace transforms be used to solve this initial value problem? Explain your answer.

Owen Jacobs
Owen Jacobs
Numerade Educator
01:52

Problem 16

Give an example of a two-dimensional nonlinear first order system for which the hypotheses of Theorem $6.1$ are not satisfied at precisely the specified points in $t y_{1} y_{2}$-space.
$$
\text { The points satisfying } 1+t+y_{1}+3 y_{2}=0
$$

AG
Ankit Gupta
Numerade Educator
00:50

Problem 17

Give an example of a two-dimensional nonlinear first order system for which the hypotheses of Theorem $6.1$ are not satisfied at precisely the specified points in $t y_{1} y_{2}$-space.
$$
\text { The points }\left(t, y_{1}, y_{2}\right)=(1, n \pi, 2), n=0, \pm 1, \pm 2, \ldots
$$

Linh Vu
Linh Vu
Numerade Educator
05:32

Problem 18

Hooke's law assumes the restoring force exerted by a spring under tension or compression is proportional to the displacement (the distance stretched or foreshortened). This assumption cannot be valid for large displacements since there are limits to the amount a spring can be stretched or compressed. Suppose we assume that the restoring force $F_{R}(x)$ is related to spring displacement $x$ by
$$
F_{R}(x)=-\frac{2 k \delta}{\pi} \tan \left(\frac{\pi x}{2 \delta}\right)
$$
In this model, the restoring force has vertical asymptotes at $x=\pm \delta$; the value $\delta$ represents the maximum amount the spring can be stretched or compressed. Consider the figure, illustrating a mass $m$ attached to such a spring. Assume that the
mass moves on a frictionless horizontal surface and that the spring has unstretched length $l$. Newton's second law of motion leads to the nonlinear differential equation
$$
m x^{\prime \prime}+\frac{2 k \delta}{\pi} \tan \left(\frac{\pi x}{2 \delta}\right)=0 .
$$
(a) Consider $\tan (\pi x / 2 \delta)$ as a function of $x$ defined on $-\delta<x<\delta$. Expand this function in a Maclaurin series. Show that if we assume $|\pi x / 2 \delta|$ is small and approximate $\tan (\pi x / 2 \delta)$ by the first nonvanishing term in this series, we obtain the linear differential equation found previously when we assumed Hooke's law to be valid.
(b) Show that if the first two nonvanishing terms of the Maclaurin expansion are retained, we obtain the differential equation
$$
m x^{\prime \prime}+k\left[x+\frac{1}{3}\left(\frac{\pi}{2 \delta}\right)^{2} x^{3}\right]=0 .
$$
Equation (14) is often used to model the onset of nonlinear effects and is referred to as modeling a spring-mass system with cubic nonlinearity.
(c) Rewrite differential equations (13) and (14) as equivalent first order systems.
(d) For each nonlinear system obtained in part (c), determine the points, if any, where the hypotheses of Theorem $6.1$ are not satisfied.

Mike Gaerlan
Mike Gaerlan
Numerade Educator
09:17

Problem 19

Nonlinear systems often arise when chemical reactions are modeled. One example is described in the reaction diagram in the figure. In the reaction shown, substance $A$ interacts reversibly with enzyme $E$ to form complex C. Complex $C$, in turn, decomposes irreversibly into the reaction product $B$ and the original enzyme $E$. The reaction rates $k_{1}, k_{1}^{\prime}$ and $k_{2}$ (assumed to be constant) are shown in the figure. With lowercase symbols used to designate concentrations, the governing differential equations are
$$
\begin{aligned}
&\frac{d a}{d t}=-k_{1} a e+k_{1}^{\prime} c \\
&\frac{d b}{d t}=k_{2} c \\
&\frac{d c}{d t}=k_{1} a e-\left(k_{1}^{\prime}+k_{2}\right) c \\
&\frac{d e}{d t}=-k_{1} a e+\left(k_{1}^{\prime}+k_{2}\right) c
\end{aligned}
$$
Typical initial conditions are $a(0)=a_{0}, b(0)=0, c(0)=0, e(0)=e_{0} .$
$$
A+E \stackrel{k_{1}^{k_{1}}}{\stackrel{k_{1}^{\prime}}{\longleftrightarrow}} C \quad C \stackrel{k_{2}}{\longrightarrow} B+E
$$
(a) Show that the differential equations (15) imply that $d[c(t)+e(t)] / d t=0$, which implies that $c(t)+e(t)=c(0)+e(0)=e_{0}$.
(b) Use the observation made in part (a) to eliminate $e(t)$ in (15) and obtain a twodimensional nonlinear system for the dependent variables $a(t)$ and $c(t)$.
(c) For the two-dimensional system obtained in part (b), at what points in tac-space are the hypotheses of Theorem $6.1$ satisfied?

Mike Gaerlan
Mike Gaerlan
Numerade Educator