Page 207 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes so that
u
x
x
W ( ) u 1 ( ) ( ) for i 1,2
x
i
i
2
i
We also know from Abel’s formula that W(x) is of one sign on x < x < x , since u (x) and u (x) are
1 2 1 2
)
x
linearly independent. This means that u (x ) and u ( ) are nonzero. Now if u (x is positive
1 i 2 i 2 1
then u 2 (x 2 ) is negative or vice versa, since u (x ) = 0. Since the Wronskian cannot change sign
2
2
between x and x , so u (x) must change sign and hence u has a zero in between x and x as we
1 2 1 1 1 2
claimed.
Self Assessment
4. Consider the equation
2
d y 2
w y 0
dx 2
It has the solution
y = A sin wx + B cos wx
If we consider any two of the zeros of sin wx, it is immediately clear that cos wx has a zero
between them.
Compare its solutions with respect to those of
2
d w 4w w 0
2
dx 2
12.7 Summary

 The comparison and separation theorems of Sturm are useful in the periodic solutions of
the second order linear equation.
 These theorems are understood in a better way once the reduction method of order is set
up.
 The variation of parameters help us in finding the particular integral of the non-
homogeneous differential equation.
12.8 Keywords

Sturm comparison theorem helps us in telling when the solution of a differential equation has at
least one zero in between the two zeros of the solution of another differential equation simply
by studying their coefficients in the equation.
Whereas, the Sturm separation theorem helps us in predicting that one independent solution of
the equation has at least one zero in between the two zeros of the other independent solution.
This happens in the case of periodic solutions.
12.9 Review Questions

x
1. Find the Wronskian of e , e x
2
d y dy
2. Find the general solution of 2 6y x
dx dx

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