Chapter 52
Power series methods
of solving ordinary
differential equations
52.1 Introduction
Second order ordinary differential equations that cannot be solved by analytical methods (as shown in
Chapters 50 and 51), i.e. those involving variable coefficients, can often be solved in the form of an infinite series
of powers of the variable. This chapter looks at some of
the methods that make this possible—by the Leibniz–
Maclaurin and Frobinius methods, involving Bessel’s
and Legendre’s equations, Bessel and gamma functions and Legendre’s polynomials. Before introducing
Leibniz’s theorem, some trends with higher differential
coefficients are considered. To better understand this
chapter it is necessary to be able to:
(i) differentiate standard functions (as explained in
Chapters 27 and 32),
(ii) appreciate the binomial theorem (as explained in
Chapters 7), and
(iii) use Maclaurins theorem (as explained in Chapter 8).
52.2 Higher order differential
coefficients as series
The following is an extension of successive differentiation (see page 296), but looking for trends, or series,
as the differential coefficient of common functions
rises.
(i) If y = e ax , then
d y
dx
= ae ax ,
d 2 y
dx 2 = a 2 e ax , and so
on.
If we abbreviate
d y
dx
as y ,
d 2 y
dx 2 as y , … and
d n y
dx n as y (n) , then y = ae ax , y = a 2 e ax , and the
emerging pattern gives: y
(n)
= a
n e
ax
(1)
For example, if y = 3e
2x , then
d 7 y
dx 7 = y (7) = 3(2 7 ) e 2x = 384e
2x
(ii) If y = sin ax,
y
= a cos ax = a sin
ax +
π
2
y
= −a
2 sin ax = a
2 sin(ax + π)
= a
2 sin
ax +
2π
2
y
= −a
3 cos x
= a
3 sin
ax +
3π
2
and so on.
Power series methods
of solving ordinary
differential equations
52.1 Introduction
Second order ordinary differential equations that cannot be solved by analytical methods (as shown in
Chapters 50 and 51), i.e. those involving variable coefficients, can often be solved in the form of an infinite series
of powers of the variable. This chapter looks at some of
the methods that make this possible—by the Leibniz–
Maclaurin and Frobinius methods, involving Bessel’s
and Legendre’s equations, Bessel and gamma functions and Legendre’s polynomials. Before introducing
Leibniz’s theorem, some trends with higher differential
coefficients are considered. To better understand this
chapter it is necessary to be able to:
(i) differentiate standard functions (as explained in
Chapters 27 and 32),
(ii) appreciate the binomial theorem (as explained in
Chapters 7), and
(iii) use Maclaurins theorem (as explained in Chapter 8).
52.2 Higher order differential
coefficients as series
The following is an extension of successive differentiation (see page 296), but looking for trends, or series,
as the differential coefficient of common functions
rises.
(i) If y = e ax , then
d y
dx
= ae ax ,
d 2 y
dx 2 = a 2 e ax , and so
on.
If we abbreviate
d y
dx
as y ,
d 2 y
dx 2 as y , … and
d n y
dx n as y (n) , then y = ae ax , y = a 2 e ax , and the
emerging pattern gives: y
(n)
= a
n e
ax
(1)
For example, if y = 3e
2x , then
d 7 y
dx 7 = y (7) = 3(2 7 ) e 2x = 384e
2x
(ii) If y = sin ax,
y
= a cos ax = a sin
ax +
π
2
y
= −a
2 sin ax = a
2 sin(ax + π)
= a
2 sin
ax +
2π
2
y
= −a
3 cos x
= a
3 sin
ax +
3π
2
and so on.