Lecture no.26. Second order linear differential equations

POINT 1. GENERAL NOTIONS

POINT 2. LINEAR DEPENDENCE AND INDEPENDENCE

POINT 3. HOMOGENEOUS EQUATIONS

POINT 4. NONHOMOGENEOUS EQUATIONS

Point 1. General notions

Def. 1. The second order linear differential equation (SO LDE, LDE) is called the next equation

. ( 1 )

Coefficients and the second [free, absolute] term of equation are known functions, is an unknown function.

Initial conditions for the equation (1) have a usual form

( 2 )

Equation (1) is called nonhomogeneous one (SO LNDE, LNDE) if its second term doesn’t equal zero identically.

If , the corresponding [associated] equation

( 3 )

is called homogeneous equation (SO LHDE, LHDE) which corresponds to the (non-homogeneous) equation (1).

Theorem 1 (on one-valued solvability of Cauchy problem). If coefficients and a second term of the equation (1) are continuous functions on some segment then Cauchy problem (1), (2) (in particular (3), (2)) has unique solution, and this solution is determined on the whole segment .

Ex. 1. Cauchy’s problem for homogeneous equation (3) with zero initial conditions

( 4 )

has unique trivial solution .

Def. 2. The left side of the differential equations (1), (3) is called a linear differential operator and is denoted by ,

. ( 5 )

The linear differential operator possesses evident properties

1. (additivity).

■

.■

2. for any constant k (homogeneity).

■ ■

As the corollary we have

for any constant (linearity).

With the help of the linear differential operator the equations (1), (3) can be written as follows

,

.

Properties of solutions of a linear homogeneous differential equation (3).

1. Sum of two solutions of the equation (3) is also a solution.

■Let be solutions of the equation (3) that is . By the property 1 of a linear differential operator .■

2. Product of any solution of the equation (3) by a constant is also a solution.

■Let y is a solution of the equation (3), i.e. , and k is a constant. Then by the property 2 of a linear differential operator .■

3. If a function is a complex solution of the equation (3) then its real and imaginary parts are also the solutions of this equation.

Point 2. Linear dependence and independence

Def. 3. Two functions are called linearly dependent on a segment if there exist two numbers not all zeros ( ) such that for any an identity

( 6 )

holds. If the identity holds only in the case , these functions are called li-nearly independent.

Theorem 2. Two functions are linearly dependent on a segment if and only if their ratio identically equals a constant on , that is for any

. ( 7 )

■1. Let functions are linearly dependent on a segment , and so the identity (5) holds for . If, for example, then we get from (6)

and a ratio of functions is an identical constant.

2. Let now on . Then

,

and are linearly dependent by the definition of linear dependence.■

Ex. 2. Functions for are linearly independent on because of their ratio isn’t a constant.

Def. 4. n functions are called linearly dependent on a segment if there exist n numbers not all zeros ( ) such that for any an identity

( 8 )

holds. If the identity holds only in the case , these functions are called linearly independent on .

Ex. 3. Functions are linearly independent on because of a polynomial in x

only if all its coefficients equal zero, i.e. .

In what follows we’ll deal with linear dependence or independence of solutions of differential equations. There is well mathematical tool to study corresponding questions namely the wronskian [Wronskian determinant, determinant of Wronski] of several functions.

Def. 5. Wronskian of functions is called the next determinant

( 9 )

Ex. 4. The wronskian of functions (see Ex. 2) is

.

Theorem 3. If functions are linearly dependent on a segment then their wronskian identically equals zero on , .

■Let for the sake of simplicity two functions are linearly dependent on .

The first mode of proving. The ratio of the functions is identical constant on . Let for example . Then the wronskian of these functions

.

The second mode of proving (it can be easily extended on any number of functions). By the definition of linear dependence there are two numbers such that and on . We differentiate this identity and compile a system of linear homogeneous equations in ,

This system has non-trivial solution, and therefore its principal determinant equals zero for any ,

.■

Theorem 4. If functions are linearly independent solutions of the nth order linear homogeneous differential equation with coefficients continuous on a segment , then the wronskian of these solutions doesn’t equal zero at all the points of the segment.

■Let’s prove the theorem for two linearly independent solutions of the second order linear homogeneous differential equation (3). Suppose that the wronskian of these functions equals zero at a point , . We choose two numbers such that and

( 10 )

It’s possible because the system (10) in has zero principal determinant . Let’s compile the next function

.

It is a solution of the equation (3) and satisfies zero initial conditions (10). Hence on the base of Ex. 1 y = 0, that is

.

But it means that the functions are linearly dependent (we remember that ).

We’ve arrived at contradiction which proves the theorem.■

Ex. 5. The functions (see Ex. 2) are linearly independent solutions of a differential equation on . Their wronskian equals (see Ex. 4) and doesn’t equal zero for any .