Download Chapter3. Series Solution of Second

Chapter3. Series Solution of Second-order differential Equations
1. Introduction
The linear differential equations we have studied so far all had closed form solutions, that is, their solutions
could be expressed in terms of elementary functions, viz. exponential, trigonometric (including inverse
trigonometric), polynomial, and logarithmic functions. As we know from calculus courses, most such
elementary functions have expansions in terms of power series. Some famous functions with their
corresponding power series are:
But there are a whole class of functions, called special functions, which are not elementary functions and
which occur frequently in mathematical physics. They usually satisfy second order homogeneous linear
differential equations. These equations can sometimes be solved by discovering a power series that satisfies
the differential equation but the solution series may not be summable to an elementary function.
2. Power Series
Let's remember some facts about the series
from calculus.

There is a nonnegative number 𝜌, called the radius of convergence, such that the series converges
absolutely for 𝑥 − 𝑥0 < 𝜌 and diverges for 𝑥 − 𝑥0 > 𝜌 . The series defines a function

in its interval of convergence.
In the interval of convergence, the series can be added or subtracted term wise, i.e.

In the interval of convergence, the series can be multiplied or divided to give another power series.
where

In the interval of convergence, derivatives and integrals of f(x) can be found by term wise
differentiation and integration, for example

The series
is called the Taylor Series of the function f(x). The function f(x)
is called analytic if its Taylor series converges.
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3. Classification of Points
Consider the equation
𝑅 𝑥 𝑦 ′′ + 𝑃 𝑥 𝑦 ′ + 𝑄 𝑥 𝑦 = 0
If both of the functions
𝑃(𝑥)
𝑄(𝑥)
,
𝑅(𝑥)
𝑅(𝑥)
are analytic at x = x0, then the point x0 is an ordinary point. Otherwise, x0 is a singular point. Suppose that x0
is a singular point of the above equation. If both of the functions
are analytic at x = x0, then the point x0 is called a regular singular point. Otherwise, x0 is an irregular singular
point. For example, the functions 1+ x + x2; sin x; ex (1+x4) cos x are all analytic at x = 0. But, the functions
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cos 𝑥 1 𝑒 𝑥 sin 𝑥 + 𝑥 + 𝑎
, , ,
𝑥 𝑥 𝑥
𝑥2
are not.
We will use power series method around ordinary points and Frobenius' method around regular singular
points. We will not consider irregular singular points.
4. Power Series Method
If x0 is an ordinary point of the equation 𝑅 𝑥 𝑦 ′′ + 𝑃 𝑥 𝑦 ′ + 𝑄 𝑥 𝑦 = 0, then the general solution is
The coefficients an can be found by inserting y in the equation and setting the coefficients of all powers to
zero. Two coefficients (Usually a0 and a1) must be arbitrary, others must be defined in terms of them. We
expect two linearly independent solutions because the equation is second order linear.
Example:
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In most applications, we want a solution close to 0, therefore we can neglect the higher order terms of the
series.
Remark: Sometimes we can express the solution in closed form (in terms of elementary functions rather
than an infinite summation) as in the next example:
Example:
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Example:
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Example:
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Example:
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Example:
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References:
1. Greenberg, M.D. Advanced Engineering Mathematics, 2nd edition. Prentice Hall, 1998.
2. O'Neil, P.V. Advanced Engineering Mathematics, 5th edition. Thomson, 2003.
3. Ross, S.L. Introduction to Ordinary Differential Equations, 4th edition. Wiley, 1989.
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