Page 204 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 12: Sturm Comparison and Separation Theorems

The variation of parameters formula then gives the particular integral as Notes
x
s
x cos sinx cos sins
y s sin s ds ,
1
since
cosx sinx
2
2
W cos x sin x 1
sinx cosx
We can split the particular integral into two integrals as
x x
2
y ( ) sinx s sin coss ds cosx s sin s ds
s
x
1 x 1 x
s
sinx s sin2s ds cosx s (1 cos2 ) ds
2 2
Using integration by parts, we can evaluate this, and find that
1 2 1 1
x
y ( ) x cosx x sinx cosx
4 4 8
is the required particular integral. The general solution is therefore
1 2 1
y c 1 cosx c 2 sinx x cosx x sinx
4 4

Self Assessment

2. Find the general solution of the equation

2
d y
4y 2sec2x
dx 2
12.4 The Wronskian

Before we carry on, let’s pause to discuss some further properties of the Wronskian. Recall that
if V is a vector space over , then two elements v , v V are linearly dependent if , ,
1 2 1 2
with and not both zero, such that v + v = 0.
1 2 1 1 2 2
1
Now let V = C (a, b) be the set of once-differentiable functions over the interval a < x < b. If u , u
1 2
1
C (a, b) are linearly dependent, ,  such that u (x) + u (x) = 0 x (a, b). Notice
1 2 1 1 2 2
that, by direct differentiation, this also gives u ( ) u ( ) 0 or, in matrix form.
x
x
1 1 2 2
x
x
u 1 ( ) u 2 ( ) 1 0
u 1 ' ( ) u ' 2 ( ) 2 0
x
x
These are homogeneous equations of the form
Ax = 0
which only have nontrivial solutions if det(A) = 0, that is
u 1 ( ) u 2 ( )
x
x
'
W u u ' u u 0.
1 2
1 2
x
u ' 1 ( ) u ' 2 ( )
x
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