Section 4.1

Initial-Value and Boundary-Value Problems

Linear Dependence and Linear Independence

Solutions of Linear Equations

INITIAL-VALUE PROBLEM

For the nth-order differential equation, the problem

,)(,,)(,)(:

)()()()()(:

)1(00

)1(0000

011

1

1

nn

n

n

nn

n

n

yxyyxyyxytoSubject

xgyxadx

dyxa

dx

ydxa

dx

ydxaSolve

where y0, y′0, etc. are arbitrary constants, is called an initial-value problem.

EXISTENCE OF A UNIQUE SOLUTION

Theorem: Let an(x), an−1(x), . . . , a1(x), a0(x), and g(x) be continuous on an interval and let an(x) ≠ 0 for every x in the interval. If x = x0 is any point in this interval, then a solution y(x) of the initial-value problem exists on the interval and is unique.

BOUNDARY-VALUE PROBLEM

A problem such as

,)(,)(:

)()()()(:

10

012

2

2

ybyyaytoSubject

xgyxadx

dyxa

dx

ydxaSolve

is called a boundary-value problem (BVP). The specified values y(a) = y0 and y(b) = y1 are called boundary conditions.

LINEAR DEPENDENCE AND LINEAR INDEPENDENCE

Definition: A set of functions f1(x), f2(x), . . . , fn(x) is said to be linearly dependent on an interval I if there exists constants c1, c2, . . . , cn, not all zero, such that

for every x in the interval.

Definition: A set of functions f1(x), f2(x), . . . , fn(x) is said to be linearly independent on an interval I if it is not linearly dependent on the interval.

0)()()( 2211 xfcxfcxfc nn

THE WRONSKIANTheorem: Suppose f1(x), f2(x), . . . , fn(x) possess at least n − 1 derivatives. If the determinant

is not zero for at least one point in the interval I, then the set of functions f1(x), f2(x), . . . , fn(x) is linearly independent on the interval.

NOTE: The determinant above is called the Wronskian of the function and is denoted by W(f1(x), f2(x), . . . , fn(x)).

)1()1(2

)1(1

21

21

nn

nn

n

n

fff

fff

fff

COROLLARYCorollary: If f1(x), f2(x), . . . , fn(x) possess at least n − 1 derivatives and are linearly dependent on I, then

W(f1(x), f2(x), . . . , fn(x)) = 0

for every x in the interval.

An linear nth-order differential equation of the form

is said to be homogeneous, whereas

g(x) not identically zero, is said to be nonhomogeneous.

HOMOGENEOUS LINEAR EQUATIONS

0)()()()( 011

1

1

yxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n

)()()()()( 011

1

1 xgyxadx

dyxa

dx

ydxa

dx

ydxa

n

n

nn

n

n

SUPERPOSITION PRINCIPLE – HOMOGENEOUS EQUATIONS

Theorem: Let y1, y2, . . . , yk be solutions of the homogeneous linear nth-order differential equation on an interval I. Then the linear combination

where the ci, i = 1, 2, . . . , k are arbitrary constants, is also a solution on the interval.

)()()( 2211 xycxycxycy kk

TWO COROLLARIES

Corollary 1: A constant multiple y = c1y1(x) of a solution y1(x) of a homogeneous linear differential equation is also a solution.

Corollary 2: A homogeneous linear differential equation always possesses the trivial solution y = 0.

CRITERION FOR LINEARLY INDEPENDENT SOLUTIONS

Theorem: Let y1, y2, . . . , yn be n solutions of the homogeneous linear nth-order differential equation on an interval I. Then the set of solutions is linearly independent on I if and only if

W(y1, y2, . . . , yn) ≠ 0

for every x in the interval.

FUNDAMENTAL SET OF SOLUTIONS

Definition: Any linearly independent sety1, y2, . . . , yn of n solutions of the homogeneous nth-order differential equation on an interval I is said to be a fundamental set of solutions on the interval.

EXISTENCE OF CONSTANTS

Theorem: Let y1, y2, . . . , yn be a fundamental set of solutions of the homogeneous linear nth-order differential equation on an interval I. Then for any solution Y(x) of the equation on I, constants C1, C2, . . . , Cn can be found so that

).()()( 2211 xyCxyCxyCY nn

EXISTENCE OF A FUNDAMENTAL SET

Theorem: There exists a fundamental set of solutions for the homogeneous linear nth-order differential equation on an interval I.

GENERAL SOLUTION – HOMOGENEOUS EQUATIONS

Definition: Let y1, y2, . . . , yn be a fundamental set of solutions of the homogeneous linear nth-order differential equation on an interval I. The general solution (or complete solution) of the equation on the interval is defined to be

where ci, i = 1, 2, . . . , n are arbitrary constants.

),()()( 2211 xycxycxycy nn

HOMEWORK

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