Differential Equations - Series Solutions: Examples - i

Before we get into finding series solutions to differential equations we need to determine when we can find series solutions to differential equations. So, let’s start with the differential equation,
$$\begin{array}{}p\left(x\right)y^{\prime \prime }+q\left(x\right)y^{\prime }+r\left(x\right)y=0& \text{(1)}\end{array}$$

This time we really do mean nonconstant coefficients. To this point we’ve only dealt with constant coefficients. However, with series solutions we can now have nonconstant coefficient differential equations. Also, in order to make the problems a little nicer we will be dealing only with polynomial coefficients.

Now, we say that $x=x_0$ is an ordinary point if provided both

$$\frac{q\left(x\right)}{p\left(x\right)}\text{and}\frac{r\left(x\right)}{p\left(x\right)}$$

are analytic at $x=x_0$. That is to say that these two quantities have Taylor series around $x=x_0$. We are going to be only dealing with coefficients that are polynomials so this will be equivalent to saying that

$$p\left(x_0\right)\ne 0$$

for most of the problems.

If a point is not an ordinary point we call it a singular point.

The basic idea to finding a series solution to a differential equation is to assume that we can write the solution as a power series in the form,

$$\begin{array}{}y\left(x\right)=\sum _{n=0}^{\infty }{a}_n{\left(x-x_0\right)}^n& \text{(2)}\end{array}$$

and then try to determine what the $a_n$’s need to be. We will only be able to do this if the point $x=x_0$, is an ordinary point. We will usually say that $\text{(2)}$ is a series solution around $x=x_0$.

Let’s start with a very basic example of this. In fact, it will be so basic that we will have constant coefficients. This will allow us to check that we get the correct solution.

Example 1 Determine a series solution for the following differential equation about $x_0=0$.
$$y^{\prime \prime }+y=0$$

Notice that in this case $p\left(x\right)=1$ and so every point is an ordinary point. We will be looking for a solution in the form,
$$y\left(x\right)=\sum _{n=0}^{\infty }{a}_n{x}^n$$We will need to plug this into our differential equation so we’ll need to find a couple of derivatives.
$$y^{\prime }\left(x\right)=\sum _{n=1}^{\infty }n{a}_n{x}^{n-1}{y}^{\prime \prime }\left(x\right)=\sum _{n=2}^{\infty }n\left(n-1\right)a_n{x}^{n-2}$$Recall from the power series review section on power series that we can start these at $n=0$ if we need to, however it’s almost always best to start them where we have here. If it turns out that it would have been easier to start them at $n=0$we can easily fix that up when the time comes around.
So, plug these into our differential equation. Doing this gives,
$$\sum _{n=2}^{\infty }n\left(n-1\right)a_n{x}^{n-2}+\sum _{n=0}^{\infty }{a}_n{x}^n=0$$The next step is to combine everything into a single series. To do this requires that we get both series starting at the same point and that the exponent on the $x$ be the same in both series.
We will always start this by getting the exponent on the $x$ to be the same. It is usually best to get the exponent to be an $n$. The second series already has the proper exponent and the first series will need to be shifted down by 2 in order to get the exponent up to an $n$. If you don’t recall how to do this take a quick look at the first review section where we did several of these types of problems.
Shifting the first power series gives us,
$$\sum _{n=0}^{\infty }\left(n+2\right)\left(n+1\right)a_{n+2}{x}^n+\sum _{n=0}^{\infty }{a}_n{x}^n=0$$Notice that in the process of the shift we also got both series starting at the same place. This won’t always happen, but when it does we’ll take it. We can now add up the two series. This gives,
$$\sum _{n=0}^{\infty }\left[\left(n+2\right)\left(n+1\right)a_{n+2}+a_n\right]x^n=0$$Now recalling the fact from the power series review section we know that if we have a power series that is zero for all $x$(as this is) then all the coefficients must have been zero to start with. This gives us the following,
$$\left(n+2\right)\left(n+1\right)a_{n+2}+a_n=0,n=0,1,2,\dots$$This is called the recurrence relation and notice that we included the values of $n$ for which it must be true. We will always want to include the values of $n$ for which the recurrence relation is true since they won’t always start at $n$ = 0 as it did in this case.
Now let’s recall what we were after in the first place. We wanted to find a series solution to the differential equation. In order to do this, we needed to determine the values of the $a_n$’s. We are almost to the point where we can do that. The recurrence relation has two different $a_n$’s in it so we can’t just solve this for $a_n$ and get a formula that will work for all $n$. We can however, use this to determine what all but two of the $a_n$’s are.
To do this we first solve the recurrence relation for the $a_n$ that has the largest subscript. Doing this gives,
$$a_{n+2}=-\frac{a_n}{\left(n+2\right)\left(n+1\right)}n=0,1,2,\dots$$Now, at this point we just need to start plugging in some value of $n$ and see what happens,

$n=0$	$a_2=\frac{-a_0}{\left(2\right)\left(1\right)}$	$n=1$	$a_3=\frac{-a_1}{\left(3\right)\left(2\right)}$
$n=2$	$\begin{array}{cc}{a}_4& =-\frac{a_2}{\left(4\right)\left(3\right)}\\ & =\frac{a_0}{\left(4\right)\left(3\right)\left(2\right)\left(1\right)}\end{array}$	$n=3$	$\begin{array}{cc}{a}_5& =-\frac{a_3}{\left(5\right)\left(4\right)}\\ & =\frac{a_1}{\left(5\right)\left(4\right)\left(3\right)\left(2\right)}\end{array}$
$n=4$	$\begin{array}{cc}{a}_6& =-\frac{a_4}{\left(6\right)\left(5\right)}\\ & =\frac{-a_0}{\left(6\right)\left(5\right)\left(4\right)\left(3\right)\left(2\right)\left(1\right)}\end{array}$	$n=5$	$\begin{array}{cc}{a}_7& =-\frac{a_5}{\left(7\right)\left(6\right)}\\ & =\frac{-a_1}{\left(7\right)\left(6\right)\left(5\right)\left(4\right)\left(3\right)\left(2\right)}\end{array}$
	$⋮$		$⋮$
	$a_{2k}=\frac{{\left(-1\right)}^k{a}_0}{\left(2k\right)!},k=1,2,\dots$		$a_{2k+1}=\frac{{\left(-1\right)}^k{a}_1}{\left(2k+1\right)!},k=1,2,\dots$

Notice that at each step we always plugged back in the previous answer so that when the subscript was even we could always write the $a_n$ in terms of $a_0$ and when the coefficient was odd we could always write the $a_n$ in terms of $a_1$. Also notice that, in this case, we were able to find a general formula for $a_n$’s with even coefficients and $a_n$’s with odd coefficients. This won’t always be possible to do.
There’s one more thing to notice here. The formulas that we developed were only for $k=1,2,\dots$ however, in this case again, they will also work for $k=0$. Again, this is something that won’t always work, but does here.
Do not get excited about the fact that we don’t know what $a_0$ and $a_1$ are. As you will see, we actually need these to be in the problem to get the correct solution.
Now that we’ve got formulas for the $a_n$’s let’s get a solution. The first thing that we’ll do is write out the solution with a couple of the $a_n$’s plugged in.
$$\begin{array}{cc}y\left(x\right)& =\sum _{n=0}^{\infty }{a}_n{x}^n\\ & =a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots +a_{2k}{x}^{2k}+a_{2k+1}{x}^{2k+1}+\cdots \\ & =a_0+a_{1}x-\frac{a_0}{2!}{x}^2-\frac{a_1}{3!}{x}^3+\cdots +\frac{{\left(-1\right)}^k{a}_0}{\left(2k\right)!}{x}^{2k}+\frac{{\left(-1\right)}^k{a}_1}{\left(2k+1\right)!}{x}^{2k+1}+\cdots \end{array}$$The next step is to collect all the terms with the same coefficient in them and then factor out that coefficient.
$$\begin{array}{cc}y\left(x\right)& =a_0\left\{1-\frac{x^2}{2!}\cdots +\frac{{\left(-1\right)}^k{x}^{2k}}{\left(2k\right)!}+\cdots \right\}+a_1\left\{x-\frac{x^3}{3!}+\cdots +\frac{{\left(-1\right)}^k}{\left(2k+1\right)!}{x}^{2k+1}+\cdots \right\}\\ & =a_0\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k{x}^{2k}}{\left(2k\right)!}+a_1\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k{x}^{2k+1}}{\left(2k+1\right)!}\end{array}$$In the last step we also used the fact that we knew what the general formula was to write both portions as a power series. This is also our solution. We are done.

Before working another problem let’s take a look at the solution to the previous example. First, we started out by saying that we wanted a series solution of the form,

$$y\left(x\right)=\sum _{n=0}^{\infty }{a}_n{x}^n$$

and we didn’t get that. We got a solution that contained two different power series. Also, each of the solutions had an unknown constant in them. This is not a problem. In fact, it’s what we want to have happen. From our work with second order constant coefficient differential equations we know that the solution to the differential equation in the last example is,

$$y\left(x\right)=c_1\cos \left(x\right)+c_2\sin \left(x\right)$$

Solutions to second order differential equations consist of two separate functions each with an unknown constant in front of them that are found by applying any initial conditions. So, the form of our solution in the last example is exactly what we want to get. Also recall that the following Taylor series,

$$\cos \left(x\right)=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n}}{\left(2n\right)!}\sin \left(x\right)=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n+1}}{\left(2n+1\right)!}$$

Recalling these we very quickly see that what we got from the series solution method was exactly the solution we got from first principles, with the exception that the functions were the Taylor series for the actual functions instead of the actual functions themselves.

Now let’s work an example with nonconstant coefficients since that is where series solutions are most useful.