Higher Engineering Mathematics, Sixth Edition

506 Higher Engineering Mathematics

thecasewhenthetwovaluesofcdifferbyaninteger(i.e.

whole number). From theabove three workedproblems,

the followingcan be deduced, and in future assumed:

(i) if two solutions of the indicial equation differ by

a quantitynotan integer, then two independent

solutionsy=u(x)+v(x)result, the general solu-

tion of which isy=Au+Bv(note: Problem 7

hadc=0and

2

3

and Problem 8 hadc=1and

1

2

;

in neither case didcdiffer by an integer)

(ii) iftwosolutionsoftheindicial equationdodifferby

an integer, as in Problem 9 wherec=0 and 1, and

if one coefficient is indeterminate, as with when

c=0, then the complete solution is always given

by using this value ofc. Using the second value

ofc,i.e.c=1 in Problem 9, always gives a series

which is one of the series in the first solution.

Now try the following exercise

Exercise 196 Further problems on power

series solution by the Frobenius method

1. Produce, using Frobenius’ method, a power
   series solution for the differential equation:

2 x

d^2 y

dx^2

+

dy

dx

−y=0.

⎡

⎢

⎢

⎢

⎢

⎢⎢

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

y=A

{

1 +x+

x^2

( 2 × 3 )

+

x^3

( 2 × 3 )( 3 × 5 )

+···

}

+Bx

1

2

{

1 +

x

( 1 × 3 )

+

x^2

( 1 × 2 )( 3 × 5 )

+

x^3

( 1 × 2 × 3 )( 3 × 5 × 7 )

+ ···

}

⎤

⎥

⎥

⎥

⎥

⎥⎥

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2. Use the Frobenius method to determine the
   general power series solution of the differen-
   tial equation:

d^2 y

dx^2

+y=0.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

y=A

(

1 −

x^2

2!

+

x^4

4!

− ···

)

+B

(

x−

x^3

3!

+

x^5

5!

− ···

)

=Pcosx+Qsinx

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

3. Determine the power series solution of the
   differential equation: 3x

d^2 y

dx^2

+ 4

dy

dx

−y= 0

using the Frobenius method.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

y=A

{

1 +

x

( 1 × 4 )

+

x^2

( 1 × 2 )( 4 × 7 )

+

x^3

( 1 × 2 × 3 )( 4 × 7 × 10 )

+···

}

+Bx−

1

3

{

1 +

x

( 1 × 2 )

+

x^2

( 1 × 2 )( 2 × 5 )

+

x^3

( 1 × 2 × 3 )( 2 × 5 × 8 )

+ ···

}

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎥⎥

⎥

⎥

⎥

⎦

4. Show, using the Frobenius method, that
   the power series solution of the differential
   equation:
   d^2 y
   dx^2

−y=0 may be expressed as

y=Pcoshx+Qsinhx,wherePandQare

constants. [Hint: check the series expansions

for coshxand sinhxon page 47]

52.6 Bessel’s equation and Bessel’s

functions

One of the most important differential equations in

applied mathematics isBessel’s equationand is of the

form:

x^2

d^2 y

dx^2

+x

dy

dx

+(x^2 −v^2 )y= 0

wherevis a real constant. The equation, which has

applications in electric fields, vibrations and heat con-

duction, may be solved using Frobenius’ method of the

previous section.

Problem 10. Determine the general power series

solution of Bessels equation.

Bessel’s equationx^2

d^2 y

dx^2

+x

dy

dx

+(x^2 −v^2 )y=0may

be rewritten as:x^2 y′′+xy′+(x^2 −v^2 )y= 0

Using the Frobenius method from page 500:

(i) Let a trial solution be of the form

y=xc{a 0 +a 1 x+a 2 x^2 +a 3 x^3 +···

+arxr+···} (34)

wherea 0 =0,