8.6. The Dimension Of The Solution Space Of A Homogeneous Linear Differential Equation

Theorem 8.4:  Dimensionality Theorem.

Let L: C ( n ) ( J )C( J )  be a linear differential operator of order n given by

        L= D n + k=1 n P k D nk          (8.7)

Then the solution space of the homogeneous equation L( y )=0  has dimension n.

Proof

Let

T:N( L ) R n   by  fT( f )=( f( x 0 ), f ( x 0 ),, f ( n1 ) ( x 0 ) )

be the linear transformation that maps each function f in the solution space N( L )  to the initial- value vector of f at x 0 J .  By the uniqueness theorem, T( f )=O  implies f=0 .  Hence, by Theorem 4.10, T is 1-1 on N( L )  so that T 1  exists and is also 1-1.  Furthermore, since every element in Rn can be an initial-value vector, the map is onto.  By Theorem 4.11, dimN( L )= R n =n .

Theorem 8.5.

Let L: C ( n ) ( J )C( J )  be a linear differential operator of order n.  Let u 1 ,, u n  be n independent solutions of the homogeneous equation L( y )=0  on J.  Then every solution y=f( x )  on J can be expressed as a linear combination

        f( x )= k=1 n c k u k ( x )                  (8.8)

where ck are constants.

Proof

This is a corollary of Theorem 8.4.

Comment

Eq(8.8) is called a general solution of L( y )=0 .

The dimensionality theorem asserts that an nth order LDE always has n independent solutions.  There is however no general method for obtaining these solutions, except for a few special types of equations, e.g., those with constant coefficients.