DIFFERENTIAL
EQUATIONS
Basic Terminology
A differential equation is an equation
that contains an unknown function together with one or more of its derivatives.
1
Examples:
1.
y 0 = 2x + cos x
2.
dy
= ky (exponential growth/decay)
dt
3.
x2y 00 − 2xy 0 + 2y = 4x3
2
4.
∂ 2u
∂ 2u
+ 2 = 0 (Laplace’s eqn.)
2
∂x
∂y
5.
d3y
d2y
dy
−4 2 +4
=0
3
dx
dx
dx
3
TYPE:
If the unknown function depends on a
single independent variable, then the
equation is an ordinary differential equation (ODE); if the unknown function
depends on more than one independent
variable, then the equation is a partial
differential equation (PDE).
4
ORDER:
The order of a differential equation is
the order of the highest derivative of
the unknown function appearing in the
equation.
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Examples:
1.
dy
= ky
dt
2.
x2y 00 − 2xy 0 + 2y = 4x3
3.
∂ 2u
∂ 2u
+ 2 =0
2
∂x
∂y
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4.
5.
d2y
dy
d3y
−4 2 +4
=0
3
dx
dx
dx
d2y
3
dy
d
xy =
2x)
+2x
sin
+3e
(e
dx2
dx
dx3
!
7
SOLUTION:
A solution of a differential equation
is a function defined on some domain
D
such that the equation reduces to
an identity when the function is substituted into the equation.
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Examples:
1.
y 0 = 2x + cos x
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2.
y 0 = ky
y = ekt
y = Cekt, C any constant
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3.
x2y 00 − 2xy 0 + 2y = 4x3
y = x2 + 2x3
Solution?
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4.
x2y 00 − 2xy 0 + 2y = 4x3
y = 2x + x2
Solution?
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4.
∂ 2u
∂ 2u
+ 2 =0
2
∂x
∂y
q
u = ln x2 + y 2
Solution?
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5.
∂ 2u
∂ 2u
+ 2 =0
2
∂x
∂y
u = cos x sinh y,
u = 3x − 4y
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6.
Find values of
r
such that
y = erx is a solution of
y 00 − 2y 0 − 15y = 0.
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7.
Find values of r such that y = xr
is a solution of
x2y 00 − 4x y 0 + 6y = 0.
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From now on, all differential equations are ordinary differential equations.
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n-PARAMETER FAMILY OF SOLUTIONS:
Example:
Solve the differential equation:
y 000 − 12x + 6e2x = 0
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Intuitively, to find a set of solutions
of an n-th order differential equation
we “integrate” n times, with each integration step producing an arbitrary
constant of integration. Thus, ”in theory,” an n-th order differential equation
has an n-parameter family of solutions.
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SOLVING A DIFFERENTIAL EQUATION:
To solve an n-th order differential equation means to find an n-parameter families of solutions. (Same n.)
20
Examples: n-parameter family of solutions:
1.
y 0 = 3x2 − 2x + 4
Answer:
y = x3 − x2 + 4x + C
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2.
y 00 = 2x + sin 2x
Answer:
1 3 1
y = x − sin 2x + C1x + C2
3
4
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3.
y 000 − 3y 00 + 3y 0 − y = 0
Answer:
y = C1ex + C2xex + C3x2ex
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4.
x2y 00 − 2xy 0 + 2y = 4x3
Answer:
y = C1x + C2x2 + 2x3
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GENERAL SOLUTION/SINGULAR
SOLUTIONS:
An “n-parameter family of solutions”
is also called the general solution.
Solutions of an n-th order differential
equation which are not included in an
n-parameter family of solutions are called
singular solutions.
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Example:
dy
= (4x + 2)(y − 2)1/3
dx
General solution:
2 + 4x + C
x
(y − 2)2/3 = 4
3
3
Singular solution: y ≡ 2
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PARTICULAR SOLUTION:
If specific values are assigned to the
arbitrary constants in the general solution of a differential equation, then the
resulting solution is called a particular
solution of the equation.
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Examples:
1.
x2y 00 − 2xy 0 + 2y = 4x3
General solution:
y = C1x + C2x2 + 2x3
Particular solutions:
y1 = 2x3
(C1 = C2 = 0)
y2 = 3x−2x2 +2x3 (C1 = 3, C2 = −2)
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dy
= (4x + 2)(y − 2)1/3
dx
General solution:
2 + 4x + C
(y − 2)1/3 = 4
x
3
3
Particular solutions:
C=0:
C = −5 :
2 + 4x
x
(y − 2)1/3 = 4
3
3
2 + 4x − 5
(y − 2)1/3 = 4
x
3
3
Singular solution: y ≡ 2
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THE DIFFERENTIAL EQUATION
OF AN n-PARAMETER FAMILY:
Given an n-parameter family of curves.
The differential equation of the family is an n-th order differential equation
that has the given family as its general
solution.
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Examples:
(1)
y 2 = Cx3 + 4
is the general solution of a first order
differential equation.
Find the equa-
tion.
Answer:
2 − 12
3y
y0 =
2xy
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2.
y = C1x + C2x3
is the general solution of a second order
equation. Find the equation.
Answer:
x2y 00 − 3xy 0 + 3y = 0
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Strategy for finding the differential
equation
Step 1.
Differentiate the family n
times. This produces a system of n + 1
equations.
Step 2.
Choose any n of the equa-
tions and solve for the parameters.
Step 3.
Substitute the “values” for
the parameters in the remaining equation.
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Examples:
The given family of functions is the
general solution of a differential equation.
(a) What is the order of the equation?
(b) Find the equation.
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1.
y = C1e2x + C2e3x + C3
(a)
(b)
35
2.
y = C1 cos 3x + C2 sin 3x
(a)
(b)
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n-th ORDER INITIAL-VALUE PROBLEMS:
1.
Find a solution of
y 0 = 3x2 + 2x + 1
which passes through the point (−2, 4).
Answer:
y = x3 + x2 + x + 10
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2.
y = C1 cos 3x + C2 sin 3x is the
general solution of
y 00 + 9y = 0.
a. Find a solution which satisfies
y(0) = 3
Answer:
y = C1 sin 3x + 3 cos 3x for
any C1.
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b. Find a solution which satisfies
y(0) = 4, y(π) = 4
Answer:
No solution!!
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c. Find a solution which satisfies
y(π/4) = 1, y 0(π/4) = 2
Answer:
5
1
√ sin 3x − √ cos 3x
3 2
3 2
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An n-th order initial-value problem
consists of an n-th order differential
equation
F x, y, y 0, y 00, . . . , y (n) = 0
together with n (initial) conditions of
the form
y(c) = k0, y 0(c) = k1, y 00(c) = k2, . . . ,
y (n−1)(c) = kn−1
where c
and k0, k1, . . . , kn−1
are
given numbers.
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NOTES:
1.
An n-th order differential equation
can always be written in the form
F x, y, y 0, y 00, · · · , y (n) = 0
by bringing all the terms to the lefthand side of the equation.
2.
The initial conditions determine
a particular solution of the differential
equation.
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Strategy for Solving an Initial-Value
Problem:
Step 1. Find the general solution of
the differential equation.
Step 2.
Use the initial conditions to
solve for the arbitrary constants in the
general solution.
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Examples:
y = C1x + C2 x3 is the general solution
of
x2y 00 − 3xy 0 + 3y = 0
a. find the solution which satisfies
y(1) = 2,
y 0(1) = −4.
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b. find the solution that satisfies
y(0) = 0,
y 0(0) = 2.
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c. find the solution that satisfies
y(0) = 2,
Answer:
y 0(0) = 3.
No solution.
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EXISTENCE AND UNIQUENESS:
The fundamental questions in a course
on differential equations are:
1.
Does a given initial-value problem
have a solution? That is, do solutions
to the problem exist ?
2.
If a solution does exist, is it
unique ? That is, is there exactly one
solution to the problem or is there more
than one solution?
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