Page 124 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 7: General Properties of Solutions of Linear Differential Equations of Order n

Now we shall prove that (ii) does not vanish at any point in D. Suppose that there exists a Notes
point x in D for which
0
y
W ( (x 0 ),y 2 (x 0 ),...,y n (x 0 )) = 0 ...(iv)
1
Then the system of linear equations with the coefficients y ( ) j (x )
i 0
C y (x 0 ) C y (x 0 ) ... C y (x 0 ) = 0
1 1
n n
2 2
C y (x ) C y (x ) ... C y (x ) = 0
1 1 0 2 2 0 n n 0
................................................................................
(n 1) (n 1) (n 1)
)
C y (x 0 ) C y (x 0 ) ... C y (x 0 = 0
1 1
2 2
n n
has solutions C , C , ...., C , not all zero. The linear combination
1 2 n
n
x
y(x) = C y ( )
i i
i 1
of y (x) with these coefficients C obviously satisfies the equation (5) and the initial conditions
i i
(6) at the point x in D. Therefore, we have
0
n
y(x) = C y i ( ) 0
x
l
i 1
x
This contradicts the fact that y 1 ( ), y 2 ( ),....,y n ( ) are linearly independent. Therefore, the
x
x
x
x
x
Wronskian of linearly independent solutions y 1 ( ), y 2 ( ),....,y n ( ) does not vanish at any
point in D.
x
x
x
Next we shall consider the Wronskian W y 1 ( ), y 2 ( ),...,y n ( ) of n solutions y (x), y (x),
1
2
x
..., y (x) where y 1 ( ),y 2 ( ),...y n ( ) are not necessarily linearly independent. Differentiating
x
x
n
x
W y 1 ( ), y 2 ( ),...,y n ( ) with respect to x, we obtain
x
x
x
x
y 1 ( ), y 2 ( ), ..., y n ( )
x
x
y ( ), y ( ), ..., y ( )
x
x
1 2 n
dW y 1 ( ),y 2 ( ),..., y n ( ) .........................................................
x
x
x
= ...(v)
dx (n 1) (n 2) (n 2)
x
x
x
y ( ), y ( ), ..., y ( )
1 2 n
n
n
x
x
y ( ) ( ), y ( ) ( ), ..., y (2) ( )
x
n
2
1
Since y (x) satisfies the equation (5)
l
n 1
x
x
x
n
y
x
y ( ) ( ) = p k ( )y (n k ) ( ) p n ( ) ( )
x
l
i
l
k 1
Substituting this in the above determinant, we obtain
dW y 1 ( ),y 2 ( ),..., y n ( )
x
x
x
= ...(vi)
dx
x
= p i ( )W y 1 ( ),y 2 ( ),...,y n ( )
x
x
x
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