9.2: Series Solutions of Linear Second Order ODEs
- Last updated
- Feb 27, 2022
- Save as PDF
- Page ID
- 98093
( \newcommand{\kernel}{\mathrm{null}\,}\)
Suppose we have a linear second order homogeneous ODE of the form
p(x)y″+q(x)y′+r(x)y=0
Suppose that p(x), q(x), and r(x) are polynomials. We will try a solution of the form
y=∞∑k=0ak(x−xo)2
and solve for the ak to try to obtain a solution defined in some interval around xo.
Definition: Ordinary and Singular Points
The point xo is called an ordinary point if p(xo)≠0 in linear second order homogeneous ODE of the form in Equation ???. That is, the functions
q(x)p(x)andr(x)p(x)
are defined for x near xo.
If p(x0)=0, then we say xo is a singular point.
Handling singular points is harder than ordinary points and so we now focus only on ordinary points.
Example 9.2.1: Expansion around an Ordinary Point
Let us start with a very simple example
y″−y=0
Let us try a power series solution near xo=0, which is an ordinary point.
Solution
Every point is an ordinary point in fact, as the equation is constant coefficient. We already know we should obtain exponentials or the hyperbolic sine and cosine, but let us pretend we do not know this.
We try
y=∞∑k=0akxk
If we differentiate, the k=0 term is a constant and hence disappears. We therefore get
y′=∞∑k=1kakxk−1
We differentiate yet again to obtain (now the k=1 term disappears)
y″=∞∑k=2k(k−1)akxk−2
We reindex the series (replace k with k+2 ) to obtain
y″=∞∑k=0(k+2)(k+1)ak+2xk
Now we plug y and y″ into the differential equation.
0=y″−y=(∞∑k=0(k+2)(k+1)ak+2xk)−(∞∑k=0akxk)=∞∑k=0((k+2)(k+1)ak+2xk−akxk)=∞∑k=0((k+2)(k+1)ak+2−ak)xk.
As y″−y is supposed to be equal to 0, we know that the coefficients of the resulting series must be equal to 0. Therefore,
(k+2)(k+1)ak+2−ak=0
or
ak+2=ak(k+2)(k+1).
The above equation is called a recurrence relation for the coefficients of the power series. It did not matter what a0 or a1 was. They can be arbitrary. But once we pick a0 and a1, then all other coefficients are determined by the recurrence relation.
Let us see what the coefficients must be. First, a0 and a1 are arbitrary
a2=a02,a3=a1(3)(2),a4=a2(4)(3)=a0(4)(3)(2),a5=a3(5)(4)=a1(5)(4)(3)(2),…
So we note that for even k, that is k=2n we get
ak=a2n=ao(2n)!
and for odd k that is k=2n+1 we have
ak=a2n+1=a1(2n+1)!
Let us write down the series
y=∞∑k=0akxk=∞∑n=0(a0(2n)!x2n+a1(2n+1)!x2n+1)=a0∞∑n=01(2n)!x2n+a1∞∑n=01(2n+1)!x2n+1.
We recognize the two series as the hyperbolic sine and cosine. Therefore,
y=aocoshx+a1sinhx
Of course, in general we will not be able to recognize the series that appears, since usually there will not be any elementary function that matches it. In that case we will be content with the series.
Example 9.2.2
Let us do a more complex example. Suppose we wish to solve Airy’s equation1, that is
y″−xy=0
near the point x0=0, which is an ordinary point.
We try
y=∞∑k=0akxk
We differentiate twice (as above) to obtain
y″=∞∑k=2k(k−1)akxk−2
We plug y into the equation
0=y″−xy=(∞∑k=2k(k−1)akxk−2)−x(∞∑k=0akxk)=(∞∑k=2k(k−1)akxk−2)−(∞∑k=0akxk+1).
We reindex to make things easier to sum
0=y″−xy=(2a2+∞∑k=1(k+2)(k+1)ak+2xk)−(∞∑k=1ak−1xk).=2a2+∞∑k=1((k+2)(k+1)ak+2−ak−1)xk.
Again y″−xy is supposed to be 0 so first we notice that a2=0 and also
(k+2)(k+1)ak+2−ak−1=0,orak+2=ak−1(k+2)(k+1).
Now we jump in steps of three. First we notice that since a2=0 we must have that, a5=0, a8=0, a11=0, etc …. In general a3n+2=0. The constants a0 and a1 are arbitrary and we obtain
a3=a0(3)(2),a4=a1(4)(3),a6=a3(6)(5)=a0(6)(5)(3)(2),a7=a4(7)(6)=a1(7)(6)(4)(3),…
For ak where k is a multiple of 3, that is k=3n we notice that
a3n=a0(2)(3)(5)(6)⋯(3n−1)(3n).
For ak where k=3n+1, we notice
a3n+1=a1(3)(4)(6)(7)⋯(3n)(3n+1).
In other words, if we write down the series for y we notice that it has two parts
y=(a0+a06x3+a0180x6+⋯+a0(2)(3)(5)(6)⋯(3n−1)(3n)x3n+⋯)=+(a1x+a112x4+a1504x7+⋯+a1(3)(4)(6)(7)⋯(3n)(3n+1)x3n+1+⋯)=a0(1+16x3+1180x6+⋯+1(2)(3)(5)(6)⋯(3n−1)(3n)x3n+⋯)=+a1(x+112x4+1504x7+⋯+1(3)(4)(6)(7)⋯(3n)(3n+1)x3n+1+⋯).
We define
y1(x)=1+16x3+1180x6+⋯+1(2)(3)(5)(6)⋯(3n−1)(3n)x3n+⋯,y2(x)=x+112x4+1504x7+⋯+1(3)(4)(6)(7)⋯(3n)(3n+1)x3n+1+⋯,
and write the general solution to the equation as y(x)=a0y1(x)+a1y2(x). Notice from the power series that y1(0)=1 and y2(0)=0. Also, y′1(0)=0 and y′2(0)=1. Therefore y(x) is a solution that satisfies the initial conditions y(0)=a0 and y′(0)=a1.

The functions y1 and y2 cannot be written in terms of the elementary functions that you know. See Figure 9.2.1 for the plot of the solutions y1 and y2. These functions have many interesting properties. For example, they are oscillatory for negative x (like solutions to y″+y=0) and for positive x they grow without bound (like solutions to y″−y=0).
Sometimes a solution may turn out to be a polynomial.
Example 9.2.3: Hermite Equation
Let us find a solution to the so-called Hermite’s equation of order n2 is the equation
y″−2xy′+2ny=0.
Find a solution around the point x0=0.
Solution
We try
y=∞∑k=0akxk.
We differentiate (as above) to obtain
y′=∞∑k=1kakxk−1,y″=∞∑k=2k(k−1)akxk−2.
Now we plug into the equation
0=y″−2xy′+2ny=(∞∑k=2k(k−1)akxk−2)−2x(∞∑k=1kakxk−1)+2n(∞∑k=0akxk)=(∞∑k=2k(k−1)akxk−2)−(∞∑k=12kakxk)+(∞∑k=02nakxk)=(2a2+∞∑k=1(k+2)(k+1)ak+2xk)−(∞∑k=12kakxk)+(2na0+∞∑k=12nakxk)=2a2+2na0+∞∑k=1((k+2)(k+1)ak+2−2kak+2nak)xk.
As y″−2xy′+2ny=0 we have
(k+2)(k+1)ak+2+(−2k+2n)ak=0,orak+2=(2k−2n)(k+2)(k+1)ak.
This recurrence relation actually includes a2=−na0 (which comes about from 2a2+2na0=0). Again a0 and a1 are arbitrary.
a2=−2n(2)(1)a0,a3=2(1−n)(3)(2)a1,a4=2(2−n)(4)(3)a2=22(2−n)(−n)(4)(3)(2)(1)a0,a5=2(3−n)(5)(4)a3=22(3−n)(1−n)(5)(4)(3)(2)a1,…
Let us separate the even and odd coefficients. We find that
a2m=2m(−n)(2−n)⋯(2m−2−n)(2m)!,a2m+1=2m(1−n)(3−n)⋯(2m−1−n)(2m+1)!.
Let us write down the two series, one with the even powers and one with the odd.
y1(x)=1+2(−n)2!x2+22(−n)(2−n)4!x4+23(−n)(2−n)(4−n)6!x6+⋯,y2(x)=x+2(1−n)3!x3+22(1−n)(3−n)5!x5+23(1−n)(3−n)(5−n)7!x7+⋯.
We then write
y(x)=a0y1(x)+a1y2(x).
We also notice that if n is a positive even integer, then y1(x) is a polynomial as all the coefficients in the series beyond a certain degree are zero. If n is a positive odd integer, then y2(x) is a polynomial. For example, if n=4, then
y1(x)=1+2(−4)2!x2+22(−4)(2−4)4!x4=1−4x2+43x4.
Footnotes
[1] Named after the English mathematician Sir George Biddell Airy (1801 – 1892).
[2] Named after the French mathematician Charles Hermite (1822–1901).