Lecture Notes

Today we begin Chapter 12, the series solution to ODE.

Consider a linear differential equation $P(x) y(x) = 0$ where $P(x)$ is some differential operator in $x$.

The philosophy is that, assume the solution exists and is analytic around $x=0$, namely, we can do Taylor series of $y(x)$ around $x=0$, then $y = \sum_{n=0}^\infty a_n x^n$ then, we can try to figure out what is the relations between $a_n$.

Examples $y' = 2x y$.

$(1-x^2) y'' - 2 xy' + l(l+1) y = 0$ where $l$ is a constant. If we plug in $y = \sum_n a_n x^n$, we get $a_{n+2} = -\frac{(l-n)(l+n+1)}{(n+2)(n+1)} a_n$ Hence, if we know $a_0$ and $a_1$, we know all the subsequence $a_i$.

The general solution is $y(x) = a_0 \left( 1 - \frac{l(l+1)}{2!}x^2 + \cdots \right) + a_1 \left( x - \frac{(l-1)(l+2)}{3!} x^3 + \cdots \right)$

We can get that, the series converges for $|x|<1$.

If $l$ is an integer, then one of the series converges. If $l_1 + l_2 = -1$, then $l_1$ and $l_2$ gives the same solution. That is why we use $l(l+1)$ to label the different solutions.

We note that, for special values of $l$, we can have a convergent solution for Legendre equation near $x^2=1$. In this example, if $l$ takes some special value (integers), then the solution space has some nice solutions (polynomial solution). Roughly speaking, such special values are called eigenvalues, and such solutions are called eigenfunctions (or eigenvectors).

$P_l(x) = \frac{1}{2^l l!} (\d_x)^l (x^2 -1)^l$ Let's show that it satisfies the Legendre equation.