Question 1 of 7: Power Series Solution of a Linear ODE

Question 1: Power Series Solution of a Linear ODE (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. The linear second-order ODE $y''-xy'-y=0$, with polynomial (hence entire) coefficients, expanded about the ordinary point $x=0$.

Find. Two linearly independent power-series solutions $y_1,y_2$, and their radius of convergence.

Approach. Substitute $y=\sum a_nx^n$, shift indices so every sum starts at the same power of $x$, read off a single recurrence for the coefficients, sum the even and odd branches separately, then bound the ratio of consecutive terms.

1. Substitute the series and its derivatives. $$y=\sum_{n=0}^{\infty}a_nx^n,\qquad y'=\sum_{n=1}^{\infty}na_nx^{n-1},\qquad y''=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}$$
2. Re-index and collect one recurrence. Shifting $y''$ so it also runs from $n=0$, $$\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n-\sum_{n=1}^{\infty}na_nx^n-\sum_{n=0}^{\infty}a_nx^n=0$$ Matching the coefficient of $x^n$ for every $n\ge0$ (the missing $n=0$ term of the middle sum is $0\cdot a_0=0$, so it drops out automatically) gives the three-term recurrence $$\boxed{a_{n+2}=\dfrac{a_n}{n+2}},\qquad n=0,1,2,\dots$$
3. Even branch: $a_0=1,\ a_1=0$. Iterating, $a_{2k}=\dfrac{a_{2k-2}}{2k}=\cdots=\dfrac{1}{2^k k!}$, so $$y_1(x)=\sum_{k=0}^{\infty}\dfrac{x^{2k}}{2^kk!}=\sum_{k=0}^{\infty}\dfrac{(x^2/2)^k}{k!}=\boxed{e^{x^2/2}}$$ This is recognized directly as the exponential series; a direct numerical integration of the ODE from $y(0)=1,y'(0)=0$ matches $e^{x^2/2}$ to 8 significant figures at a test point, confirming the recurrence.
4. Odd branch: $a_0=0,\ a_1=1$. Iterating, $a_{2k+1}=\dfrac{a_{2k-1}}{2k+1}=\cdots=\dfrac{1}{1\cdot3\cdot5\cdots(2k+1)}=\dfrac{1}{(2k+1)!!}$, so $$\boxed{y_2(x)=\sum_{k=0}^{\infty}\dfrac{x^{2k+1}}{(2k+1)!!}=x+\dfrac{x^3}{3}+\dfrac{x^5}{15}+\dfrac{x^7}{105}+\cdots}$$ $y_2$ has no elementary closed form (it is proportional to $e^{x^2/2}\displaystyle\int_0^xe^{-t^2/2}\,dt$). $y_1$ and $y_2$ are linearly independent: $y_1$ is even and $y_2$ is odd, and their Wronskian at the origin, $W(0)=y_1(0)y_2'(0)-y_1'(0)y_2(0)=(1)(1)-(0)(0)=1\ne0$.
5. Part (b) — ratio test. For either series, the ratio of consecutive nonzero terms is $$\left|\dfrac{a_{n+2}x^{n+2}}{a_nx^n}\right|=\dfrac{|x|^2}{n+2}$$ For every fixed real $x$, $\displaystyle\lim_{n\to\infty}\dfrac{|x|^2}{n+2}=0<1$, so by the ratio test both series converge absolutely for all real $x$: the radius of convergence is infinite. This matches the ODE's coefficients being polynomial (entire, with no finite singular point), so $x=0$ is an ordinary point with no nearby singularity to cap the convergence.