Page 205 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes In other words, the Wronskian of two linearly dependent functions is identically zero on (a, b).
The contrapositive of this result is that if W 0 on (a, b), then u and u are linearly independent
1 2
on (a, b).

2
3
Example 1: The functions u (x) = x and u (x) = x are linearly independent on the interval
1 2
3
2
( 1, 1). To see this, note that, since u (x) = x , u (x) = x , u ( ) 2 , and u ( ) 3x 2 , the Wronskian
x
x
x
1 2 1 2
of these two functions is
x 2 x 3
W 3x 4 2x 4 x 4
2x 3x 2
2
3
This quantity is not identically zero, and hence x and x are linearly independent on ( 1, 1)
Example 2: The functions u (x) = f(x) and u (x) = kf(x), with k a constant, are linearly
1 2
dependent on any interval, since their Wronskian is
f kf
W 0
f ' kf '
If the functions u and u are solutions of (2), we can show by differentiating W u u u u
1 2 1 2 1 2
directly that

dW
x
a 1 ( )W 0.
dx
This first order differential equation has solution
x
t
W ( ) W (x 0 )exp a 1 ( )dt ...(7)
x
0 x
which is known as Abel’s formula. This gives us an easy way of finding the Wronskian of the
solutions of any second order differential equation without having to construct the solutions
themselves.

Example 3: Consider the equation

1 1
y y 1 y 0
x x 2
Using Abel’s formula, this has Wronskian

x dt x W (x ) A
W ( ) W (x 0 )exp 0 0
x
0 x t x x
for some constant A.
We end this section with a useful theorem.
Theorem. If u and u are linearly independent solutions of the homogeneous, non-singular
1 2
ordinary differential equation (2), then the Wronskian is either strictly positive or strictly negative.
Proof: From Abel’s formula, and since the exponential function does not change sign, the
Wronskian is identically positive, identically negative or identically zero. We just need to

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