8.11.1. Method Of Variation Of Parameters

Given n independent solutions u 1 ,, u n  of the nth order homogeneous equation L( y )=0 , the method of variation of parameters provides a particular solution to the nonhomogeneous equation L( y )=R  of the form

        y 1 = k=1 n ν k u k                              (8.17)

where nk are functions to be determined.

 

In terms of vector notations, (8.17) can be written as an inner product

        y 1 =( ν,u )   =u|ν           (8.18)

where

        ν=( v 1 ,, v n )   and  u=( u 1 ,, u n )

The derivative of the vector- valued function n is defined to be

        ν =( v 1 ,, v n )

The Leibniz rule ( fg ) = f g+f g  also applies to the inner product.  Thus,

        y 1 =( ν ,u )+( ν, u )                   (8.19)

 

The nk ’s are to be determined from the following n conditions,

        ( ν , u ( k ) )=0              for k=0,1,,n2                         (8.19a)

        ( ν , u ( n1 ) )=R                                                                   (8.19b)

where u ( 0 ) =u .  The meaning of these conditions are as follows: 

Using (8.19a), the 1st n-1 derivatives of y 1  become

        y 1 =( ν ,u )+( ν, u ) =( ν, u )

        y 1 =( ν , u )+( ν, u ) =( ν, u )

                     

        y 1 ( n1 ) =( ν , u ( n2 ) )+( ν, u ( n1 ) ) =( ν, u ( n1 ) )

With the adiitional help of (8.19b), the nth derivative becomes

        y 1 ( n ) =( ν , u ( n1 ) )+( ν, u ( n ) )   =R+( ν, u ( n ) )

 

Thus, if

        L= D n + k=1 n P k ( x ) D nk

where D 0 =1 , we have

        L( y 1 )= y 1 ( n ) + k=1 n P k ( x ) y 1 ( nk )   =R+( ν, u ( n ) )+ k=1 n P k ( x )( ν, u ( nk ) )

                =R+( ν,L( u ) )   =R

where we have used L( u )=O .  Therefore, the conditions (8.19a,b) will make y1 of (8.17) a particular solution of the nonhomogeneous equation.  With the help of the Wronskian matrix of u 1 ,, u n , i.e.,

        W=( u 1 u 2 u n u 1 u 2 u n u 1 ( n1 ) u 2 ( n1 ) u n ( n1 ) )

eqs(8.19a,b) can be thrown into the matrix form

        W( x ) V ( x )=R( x ) E n             where                      (8.20)

where V=( ν 1 v n )  is the column matrix version of n and E n =( 0 0 1 )  is that of the nth unit coordinate vector.  As a check, the ith component of (8.20) is

        k=1 n W ik V k = k=1 n u k ( i1 ) ν k =( ν , u ( i1 ) )=R δ in

 

In §8.12, we shall show that W is nonsingular.  Hence, W 1  exists and (8.20) becomes

        V ( x )=R( x ) W 1 ( x ) E n          (8.20a)

 

Integrating over an interval [ c,x ]J , we have

        V( x )V( c )= c x dt   R( t ) W 1 ( t ) E n   Z( x )

Hence,

        V( x )=V( c )+Z( x )

so that (8.18) becomes

        y 1 =( u,v ) =( u,  ν( c )+z( x ) )   =( u,ν( c ) )+( u,z( x ) )

where z is the vector whose column matrix form is Z.  Now, ( u,ν( c ) )  is a linear combination of the uk’s with constant coefficients ν k ( c ) .  Therefore, L( u,ν( c ) )=O .  Thus, the particular solution is given by

        y 1 =( u,z( x ) )   where  Z( x )= c x dt   R( t ) W 1 ( t ) E n

Thus, we have proved the following theorem.

Theorem 8.11.

Given n independent solutions u 1 ,, u n  of the nth order homogeneous equation L( y )=0  on an interval J.  Then a particular solution to the nonhomogeneous equation L( y )=R  is

        y 1 = k=1 n ν k u k

where the nk ’s are the components of the column matrix V given by

        V( x )= c x dt   R( t ) W 1 ( t ) E n

where c,xJ .