SOLUTION MANUAL
All Chapters Included
, Solutions Manual To Accompany Ordinary Differential Equations
TABLE OF CONTENT
Page
Chapter 1 First-Order Differential Equations 1
2 Higher-Order Linear Equations 34
3 Applications of Higher-Order Equations 56
4 Systems of Linear Differential Equations 70
5 Laplace Transform 107
6 Series Solutions 127
7 Systems of Nonlinear Differential Equations 148
CAS Tutorial 167
1. Maple 167
2. MATLAB 172
3. Mathematica 176
4. Maple for this Text, by Chapter 180
,v
, CHAPTER 1
First-Order Differential Equations
Section 1.1
This first section is simply to introduce you to differential equations: what they look like, some
ideas as to how they arise in applications, and some important definitions. We see that the complete
problem might be not just the differential equation, but also one or more "initial conditions." If such
conditions are prescribed, the problem is called an initial value problem, or IVP. For instance, (6)
[that is, equation (6) in the text] is an IVP because in addition to the DE (differential equation) there
are two initial conditions, given by (6b), so that the solution of the IVP must satisfy not only the DE
(6a), but also those two initial conditions.
Chapter 1 is about first-order equations; that is, equations in which the highest derivative is of
first order. In that case, hence all through Chapter 1, there will be only one initial condition. In later
chapters we will find that the "appropriate" number of initial conditions for a DE is the same as the
order of the equation. For instance, (6a) is of second order and, sure enough, there are two initial
conditions in (6b).
The distinction between linear and nonlinear differential equations will be of great importance,
so it is necessary to be able to tell if a given equation is linear or nonlinear. Later, we will find that
the key is whether or not a certain linearity property is satisfied, but for now it will suffice not to
know about that property, but simply to say that an nth-order equation is linear if it is in, or can be put
into, the form (14). What is the form of (14)? First, put all occurrences of the unknown, that is, the
dependent variable such as y in (14), on the LHS (left-hand side of the equation); anything else goes
on the right. If the LHS is a linear combination of then the DE is linear. Actually, the
"constants" that multiply in (14) are permitted to be functions of x; the point is that they
don't depend on or its derivatives.
EXAMPLES
Example 1. (Definitions) State the order of the
,
whether it is linear or nonlinear, homogeneous or nonhomogeneous, and determine whether or not the
given functions are solutions, that is, whether or not they "satisfy" the DE:
SOLUTION. The equation is of second order because the highest derivative present is of second
order; it is linear because it is of the form (14), with and
and it is nonhomogeneous because the RHS, , is not zero. The RHS does happen to
be 0 at ,
All Chapters Included
, Solutions Manual To Accompany Ordinary Differential Equations
TABLE OF CONTENT
Page
Chapter 1 First-Order Differential Equations 1
2 Higher-Order Linear Equations 34
3 Applications of Higher-Order Equations 56
4 Systems of Linear Differential Equations 70
5 Laplace Transform 107
6 Series Solutions 127
7 Systems of Nonlinear Differential Equations 148
CAS Tutorial 167
1. Maple 167
2. MATLAB 172
3. Mathematica 176
4. Maple for this Text, by Chapter 180
,v
, CHAPTER 1
First-Order Differential Equations
Section 1.1
This first section is simply to introduce you to differential equations: what they look like, some
ideas as to how they arise in applications, and some important definitions. We see that the complete
problem might be not just the differential equation, but also one or more "initial conditions." If such
conditions are prescribed, the problem is called an initial value problem, or IVP. For instance, (6)
[that is, equation (6) in the text] is an IVP because in addition to the DE (differential equation) there
are two initial conditions, given by (6b), so that the solution of the IVP must satisfy not only the DE
(6a), but also those two initial conditions.
Chapter 1 is about first-order equations; that is, equations in which the highest derivative is of
first order. In that case, hence all through Chapter 1, there will be only one initial condition. In later
chapters we will find that the "appropriate" number of initial conditions for a DE is the same as the
order of the equation. For instance, (6a) is of second order and, sure enough, there are two initial
conditions in (6b).
The distinction between linear and nonlinear differential equations will be of great importance,
so it is necessary to be able to tell if a given equation is linear or nonlinear. Later, we will find that
the key is whether or not a certain linearity property is satisfied, but for now it will suffice not to
know about that property, but simply to say that an nth-order equation is linear if it is in, or can be put
into, the form (14). What is the form of (14)? First, put all occurrences of the unknown, that is, the
dependent variable such as y in (14), on the LHS (left-hand side of the equation); anything else goes
on the right. If the LHS is a linear combination of then the DE is linear. Actually, the
"constants" that multiply in (14) are permitted to be functions of x; the point is that they
don't depend on or its derivatives.
EXAMPLES
Example 1. (Definitions) State the order of the
,
whether it is linear or nonlinear, homogeneous or nonhomogeneous, and determine whether or not the
given functions are solutions, that is, whether or not they "satisfy" the DE:
SOLUTION. The equation is of second order because the highest derivative present is of second
order; it is linear because it is of the form (14), with and
and it is nonhomogeneous because the RHS, , is not zero. The RHS does happen to
be 0 at ,
