differential equation

492 DIFFERENTIAL EQUATIONS
For example, if y = 4 cos 2x,
(
then d6 y
dx 6 = y(6) = 4(2 6 ) cos 2x + 6π )
2
= 4(2 6 ) cos(2x + 3π)
= 4(2 6 ) cos(2x + π)
= −256 cos 2x
(iv) If y = x a , y ′ = ax a−1 , y ′′ = a(a − 1)x a−2 ,
y ′′′ = a(a − 1)(a − 2)x a−3 ,
and y (n) = a(a−1)(a−2) .....(a−n+1) x a−n
or y (n) a!
=
(a − n)! xa−n (4)
where a is a positive integer.
For example, if y = 2x 6 , then d4 y
dx 4 = y(4)
6!
= (2)
(6 − 4)! x6−4
= (2) 6 × 5 × 4 × 3 × 2 × 1
2 × 1
= 720x 2
(v) If y = sinh ax,
y ′ = a cosh ax
y ′′ = a 2 sinh ax
x 2
y ′′′ = a 3 cosh ax, and so on
Since sinh ax is not periodic (see graph on page
43), it is more difficult to find a general statement
for y (n) . However, this is achieved with the
following general series:
y (n) = an
2 {[1 + (−1)n ] sinh ax
For example, if
+ [1 − (−1) n ] cosh ax} (5)
y = sinh 2x, then d5 y
dx 5 = y(5)
= 25
2 {[1 + (−1)5 ] sinh 2x
+ [1 − (−1) 5 ] cosh 2x}
= 25
{[0] sinh 2x + [2] cosh 2x}
2
= 32 cosh 2x
(vi) If y = cosh ax,
y ′ = a sinh ax
y ′′ = a 2 cosh ax
y ′′′ = a 3 sinh ax, and so on
Since cosh ax is not periodic (see graph on page
43), again it is more difficult to find a general
statement for y (n) . However, this is achieved
with the following general series:
y (n) = an
2 {[1 − (−1)n ] sinh ax
+ [1 + (−1) n ] cosh ax} (6)
For example, if y = 1 cosh 3x,
9) then d7 y 1 3
7
=(
dx 7 = y(7) (2 sinh 3x)
9 2
= 243 sinh 3x
(vii) If y = ln ax, y ′ = 1 x , y′′ =− 1 x 2 , y′′′ = 2 x 3 , and
so on.
In general, y (n) n−1 (n − 1)!
= (−1)
x n (7)
For example, if y = ln 5x, then
d 6 ( )
y
5!
dx 6 = y(6) = (−1) 6−1 x 6 = − 120
x 6
Note that if y = ln x, y ′ = 1 ; if in equation (7),
x
n = 1 then y ′ = (−1) 0 (0)!
x 1
(−1) 0 = 1 and if y ′ = 1 then (0)! = 1 (Check
x
that (−1) 0 = 1 and (0)! =1 on a calculator).
Now try the following exercise.
Exercise 194 Further problems on higher
order differential coefficients as series
Determine the following derivatives:
1. (a) y (4) when y = e 2x (b) y (5) when y = 8e t 2
[(a) 16 e 2x (b) 1 4 e t 2 ]
2. (a) y (4) when y = sin 3t
(b) y (7) when y = 1 sin 5θ
50
[(a) 81 sin 3t (b) −1562.5 cos 5θ]