Higher Engineering Mathematics

494 DIFFERENTIAL EQUATIONS

Differentiating each term ofx^2 y′′+ 2 xy′+y= 0
ntimes, using Leibniz’s theorem of equation (13),
gives:
{
y(n+2)x^2 +ny(n+1)(2x)+

n(n−1)

2!

y(n)(2)+ 0

}

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

i.e. x^2 y(n+2)+ 2 nxy(n+1)+n(n−1)y(n)

+ 2 xy(n+1)+ 2 ny(n)+y(n)= 0

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

+(n^2 −n+ 2 n+1)y(n)= 0

or x^2 y(n+2)+ 2 (n+ 1 )xy(n+1)

+(n^2 +n+1)y(n)= 0

Problem 3. Differentiate the following

differential equationntimes:

(1+x^2 )y′′+ 2 xy′− 3 y= 0

By Leibniz’s equation, equation (13),
{

y(n+2)(1+x^2 )+ny(n+1)(2x)+

n(n−1)

2!

y(n)(2)+ 0

}

+ 2 {y(n+1)(x)+ny(n)(1)+ 0 }− 3 {y(n)}= 0

i.e. (1+x^2 )y(n+2)+ 2 nxy(n+1)+n(n−1)y(n)

+ 2 xy(n+1)+ 2 ny(n)− 3 y(n)= 0

or (1+x^2 )y(n+2)+2(n+1)xy(n+1)

+(n^2 −n+ 2 n−3)y(n)= 0

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

+(n^2 +n−3)y(n)= 0

Problem 4. Find the 5th derivative of

y=x^4 sinx

Ify=x^4 sinx, then using Leibniz’s equation with

u=sinxandv=x^4 gives:

y(n)=

[

sin

(

x+

nπ

2

)

x^4

]

+n

[

sin

(

x+

(n−1)π

2

)

4 x^3

]

+

n(n−1)

2!

[

sin

(

x+

(n−2)π

2

)

12 x^2

]

+

n(n−1)(n−2)

3!

[

sin

(

x+

(n−3)π

2

)

24 x

]

+

n(n−1)(n−2)(n−3)

4!

[

sin

(

x

+

(n−4)π

2

)

24

]

and y(5)=x^4 sin

(

x+

5 π

2

)

+ 20 x^3 sin (x+ 2 π)

+

(5)(4)

2

(12x^2 ) sin

(

x+

3 π

2

)

+

(5)(4)(3)

(3)(2)

(24x) sin(x+π)

+

(5)(4)(3)(2)

(4)(3)(2)

(24) sin

(

x+

π

2

)

Since sin

(

x+

5 π

2

)

≡sin

(

x+

π

2

)

≡cosx,

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)+120 cosx

i.e. y(^5 )=(x^4 − 120 x^2 + 120 )cosx

+( 20 x^3 − 240 x)sinx

Now try the following exercise.

Exercise 195 Further problems on 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)(6x)+n(n−1)(n−2)}

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

⎤

⎥

⎦