Existence of a Unique Solution

Let the coefficient functions and g(x) be continuous on an interval I and let the leading coefficient function not equal 0 for every x in I. If x is any point in this interval, then a solution y(x) of the initial-value problem exists on the interval and is unique.

Homogeneous EquationsNonhomogeneous Equations

1

1 1 01

1

1 1 01

( ) ( ) .......... ( ) 0

( ) ( ) .......... ( ) ( )

n n

n nn n

n n

n nn n

d y d y dya x a x a a x y

dx dx dx

d y d y dya x a x a a x y g x

dx dx dx

Superposition Principle—Homogeneous Equations

Let be solutions of the homogeneous nth-order differential equation on an interval I. Then the linear combination

where the coefficients are arbitrary constants, is also a solution on the interval.

1 2, , ...... ky y y

1 1 2 2( ) ( ) ....... ( )k ky c y x c y x c y x

Corollaries

• A constant multiple of a solution of a homogeneous linear differential equation is also a solution.

• A homogeneous linear differential equation always possesses the trivial solution .

1 1( )y c y x

0y

Linear Dependence/Independence

A set of functions is said to be linearly dependent on an interval I if there exist constants , not all zero, such that for every x in the interval. If the set of functions is not linearly dependent on the interval, it is said to be linearly independent.

1 2( ), ( ), ....... ( )nf x f x f x

1 2, ,....... nc c c

1 1 2 2( ) ( ) ...... ( ) 0n nc f x c f x c f x

Criterion for Linearly Independent Solutions

Let 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 for every x in the interval.

1 2, ,......, ny y y

1 2( , ,....., ) 0nW y y y

Fundamental Set of Solutions

Let set of n linearly independent solutions of the homogeneous linear nth-order differential equation on an interval I is said to be a fundamental set of solutions on the interval.

There exists a fundamental set of solutions for the homogeneous linear nth-order DE on an interval I.

1 2, ,......., ny y y

General Solution-Homogeneous Equation

Let be a fundamental set of solutions of the homogeneous linear nth-order differential equation on an interval I. Then the general solution of the equation on the interval is

where the coefficients are arbitrary constants.

1 2, ,......., ny y y

1 1 2 2( ) ( ) ....... ( )n ny c y x c y x c y x