Higher Engineering Mathematics, Sixth Edition

Power series methods of solving ordinary differential equations 497

sin(x+ 2 π)≡sinx,sin

(

x+

3 π

2

)

≡−cosx,

and sin(x+π)≡−sinx,

then y(^5 )=x^4 cosx+ 20 x^3 sinx+ 120 x^2 (−cosx)

• 240 x(−sinx)+120cosx

i.e. y(5)=(x^4 − 120 x^2 +120)cosx

+(20x^3 − 240 x)sinx

Now try the following exercise

Exercise 194 Further problemson

Leibniz’s theorem

Use the theorem of Leibniz in the following

problems:

1. Obtain then’th derivative of:x^2 y.
   [
   x^2 y(n)+ 2 nxy(n−^1 )+n(n− 1 )y(n−^2 )

]

2. Ify=x^3 e^2 xfindy(n)and hencey(^3 ).
   ⎡
   ⎢
   ⎣

y(n)=e^2 x 2 n−^3 { 8 x^3 + 12 nx^2

+n(n− 1 )( 6 x)+n(n− 1 )(n− 2 )}

y(^3 )=e^2 x( 8 x^3 + 36 x^2 + 36 x+ 6 )

⎤

⎥

⎦

3. Determine the 4th derivative of:y= 2 x^3 e−x.
   [y(^4 )=2e−x(x^3 − 12 x^2 + 36 x− 24 )]


4. Ify=x^3 cosxdetermine the 5th derivative.
   [y(^5 )=( 60 x−x^3 )sinx+
   ( 15 x^2 − 60 )cosx]


5. Find an expression fory(^4 )ify=e−tsint.
   [y(^4 )=−4e−tsint]


6. Ify=x^5 ln2xfindy(^3 ).
   [y(^3 )=x^2 ( 47 +60ln2x)]


7. Given 2x^2 y′′+xy′+ 3 y=0 show that
   2 x^2 y(n+^2 )+( 4 n+ 1 )xy(n+^1 )+( 2 n^2 −n+
   3 )y(n)=0.


8. Ify=(x^3 + 2 x^2 )e^2 xdetermine an expansion
   fory(^5 ).
   [y(^5 )=e^2 x 24 ( 2 x^3 + 19 x^2 + 50 x+ 35 )]

52.4 Power series solution by the

Leibniz–Maclaurin method

For second order differential equations that cannot be

solved by algebraic methods, theLeibniz–Maclaurin

methodproduces a solution in the form of infinite

series of powers of the unknown variable. The fol-

lowing simple5-step proceduremaybeusedinthe

Leibniz–Maclaurin method:

(i) Differentiate the given equation n times, using

the Leibniz theorem of equation (13),

(ii) rearrange the result to obtain the recurrence

relation atx=0,

(iii) determine the values of the derivatives atx=0,

i.e. find(y) 0 and(y′) 0 ,

(iv) substitute in the Maclaurin expansion for

y=f(x)(see page 69, equation (5)),

(v) simplify the result where possible and apply

boundary condition (if given).

The Leibniz–Maclaurin method is demonstrated, using

the above procedure, in the followingworked problems.

Problem 5. Determine the power series solution

of the differential equation:

d^2 y

dx^2

+x

dy

dx

+ 2 y=0 using Leibniz–Maclaurin’s

method, given the boundary conditions that at

x=0,y=1and

dy

dx

=2.

Following the above procedure:

(i) The differential equation is rewritten as:

y′′+xy′+ 2 y=0 and from the Leibniz theorem

of equation (13), each term is differentiatedn

times, which gives:

y(n+^2 )+{y(n+^1 )(x)+ny(n)( 1 )+ 0 }+ 2 y(n)= 0

i.e. y(n+^2 )+xy(n+^1 )+(n+ 2 )y(n)= 0

(14)

(ii) Atx=0, equation (14) becomes:

y(n+^2 )+(n+ 2 )y(n)= 0

from which, y(n+^2 )=−(n+ 2 )y(n)