C:\Users\torok\Desktop\3321\Ch1-slides_notes.xopp2 First Order Differential Equations and Appli-
cations
4 The Laplace Transform
Appendices: Appendix-1 Complex Numbers Appendix-2 Polynomials Appendix-3 Tables
DIFFERENTIAL EQUATIONS
1.1 Preliminaries
• Real numbers and intervals
terval then . . .
– Thm. 3: Mean Value Theorem
– Corollaries
1
• Integration:
f(x)dx,
b
– Thm. 5: Fundamental Thm. of Calcu-
lus
– f ± g,
f · g,
f/g, etc.
1.2. Basic Terminology
gether with one or more of its deriva-
tives.
2
Examples:
2. dy
3
5. d3y
dx3 − 4
single independent variable, then the
equation is an
more than one independent variable,
then the equation is a
partial differential equation (PDE).
the unknown function appearing in
the equation.
2. dy
7
5. d3y
dx3 − 4
is a function defined on some domain
D such that the equation reduces to
an identity when the function is substi-
tuted into the equation.
10
Is y = 2e4x − 1 2 e2x a solution?
12
13
Is y = 3 2 x4 + 2x3 a solution?
14
15
16
∂2u
∂x2 +
∂2u
Solutions??
17
y − 3y − 10y = 0.
x2y +2x y − 6y = 0.
19
y − 1
tions are ordinary differential equa-
tions.
21
tion:
tion having the special form
y(n)(x) = f(x),
an arbitrary constant;
trary constants.
an n-th order differential equation
F x, y, y, y, . . . , y(n)
= 0
tegration step producing an arbitrary
constant of integration (i.e., a param-
eter). Thus, ”in theory,” an n-th order
differential equation has an n-parameter
family of solutions.
tion
= 0
solutions. (Note: Same n.)
solutions” is more commonly called the
GENERAL SOLUTION. 25
1. y − 3x2 − 2x+4 = 0
26
27
28
29
Answer: y = C1e x + C2xe
x + C3x 2ex
3 + 3 2 x4
arbitrary constants in the general solu-
tion of a differential equation, then the
resulting solution is called a particular
solution of the equation.
Particular solutions:
General solution:
Particular solutions:
33
The correct way to give the additional information (in order to determine the parameters) is called an Initial Value Problem (IVP), see later.
THE DIFFERENTIAL EQUATION
ily is an n-th order differential equation
that has the given family as its general
solution.
34
Examples:
tion of a DE.
b. Find the DE?
solution of a DE.
b. Find the DE?
ferential equation
times. This produces a system of n+1
equations.
tions and solve for the parameters.
Step 3. Substitute the “values” for
the parameters in the remaining equa-
tion. 37
general solution of a differential equa-
tion.
(b) Find the equation.
3x
(a)
(b)
40
3. y = C1 cos 3x+ C2 sin 3x
(a)
(b)
41
(a)
(b)
42
(a)
(b)
43
y = 3x2 + 2x+1
44
how to get a particular solution from the general solution
a particular solution
the general solution
tion)
45
solution that satisfies the
2. y = C1 cos 3x + C2 sin 3x is the
general solution of
y +9y = 0.
y(0) = 3
b. Find a solution which satisfies
y(0) = 3, y(0) = 4
48
y = 3cos 3x + 4 3 sin 3x is the solution
of
49
y(0) = 4, y(π) = 4
y(0) = 4, y(π) = −4
consists of an n-th order differential
equation
= 0
the form
y(n−1)(c) = kn−1
given numbers.
can always be written in the form
F x, y, y, y, · · · , y(n)
= 0
hand side of the equation.
2. The initial conditions determine
a particular solution of the differential
equation.
52
Problem:
the differential equation.
general solution.
problem
54
4x is the general
conditions
55
solution of
y − 3
x y +
y(1) = 2, y(1) = −4.
56
y(0) = 0, y(0) = 2.
c. Find a solution which satisfies
y(0) = 4, y(0) = 3.
57
WHAT IS WRONG? the ODE is not defined at x=0!
EXISTENCE AND UNIQUENESS:
on differential equations are:
have a solution? That is, do solutions
to the problem exist ?
unique ? That is, is there exactly one
solution to the problem or is there more
than one solution? 58