Page 65 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
On the infinite open interval ( , ). Here we take as boundary conditions the following: as
x , and as x , ( ) tends to infinity of an order not greater than a certain finite power
x
y
of x, i.e.
y(x) = 0 x k as x ...(B)

The equation (i) is written as
2
2 d y 2 dy 2
e x 2xe x 2 e x y = 0
dx 2 dx
or
2
d y 2x dy 2 y = 0
dx 2 dx ...(i)

dy
From the coefficients of and y, it is clear that there are no singular points except x .
dx
Hence its solution can be given by a power series by Frobenius method

y(x) = a x r k ...(ii)
r
r 0
Which converges for x .

dy
= (r k )a x k r 1
dx r
r 0
2
d y k r 2
and 2 = a r (k r )(k r 1)x
dx
r 0
Substituting in (i), we have

a r (k r )(k r 1)x k r 2 2(k r )x k r 2 x k r = 0,
r 0

or a r (k r )(k r 1)x k r 2 2(k r )x k r = 0 ...(iii)
r 0
Now (iii) being an identity, we can equate to zero the coefficients of various powers of x.

Equating to zero the coefficient of lowest power of x, i.e., of x k 2 , we get
a k(k 1) = 0.
0
Now a 0, as it is the coefficient of the first term with which the series is started.
0
either k = 0
or k = 1 ...(iv)
k 1
Equating the coefficient of x in (iii) to zero, we get
a 1 (k 1)k = 0 ...(v)

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