Chapter 4.2, Problem 37P

The differential equation

$xy"-(x+N)y\text{'}+Ny=0,$

Where N is nonnegative integer, has been discussed by several authors. One reason why it is interesting is that it has an exponential solution and polynomial solution.

(a) Verify that one solution is $y_1(x)=e^x$.

(b) Show that a second solution has the form $y_2(x)=c{e}^x{\displaystyle \int x^N{e}^{-x}dx.}$ Calculate $y_2(x)$ for $N=1$ and $N=2$; Convince yourself that, with $c=\frac{-1}{N!},$

$y_2(x)=1+\frac{x}{1!}+\frac{x^2}{2!}+\text{}\mathrm{......}+\text{}\frac{x^n}{n!}$

Note that $y_2(x)$ is exactly the first $N+1$ term in the Taylor series about $x=0$ for $e^x$, that is,

for $y_1(x)$.