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Here's a question I'm struggling with and I'd like help on it. Please, this is not a homework problem.

I want to find the power series solutions about the origin of two linearly independent solutions of $$w''-zw=0.$$

Also, how do I show that these solutions are analytic?

thanks.

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2  
Look up the Airy function. –  J. M. Feb 16 '12 at 6:47
    
For complex functions, once differentiable implies analytic. So you can assume power series... –  Aryabhata Feb 16 '12 at 7:17

2 Answers 2

up vote 2 down vote accepted

$w(x)=a_0+a_1x+\frac{a_2x^{2}}{2!}+\frac{a_3x^{3}}{3!}+...$

$w''(x)=a_2+a_3x+\frac{a_4x^{2}}{2!}+\frac{a_5x^{3}}{3!}+...$

$w''-xw=0$

$(a_2+a_3x+\frac{a_4x^{2}}{2!}+\frac{a_5x^{3}}{3!}+...)-x(a_0+a_1x+\frac{a_2x^{2}}{2!}+\frac{a_3x^{3}}{3!}+...)=0$

$a_2+(a_3-a_0)x+(\frac{a_4}{2!}-a_1)x^{2}+(\frac{a_5}{3!}-\frac{a_2}{2!})x^{3}+....=0$

$a_2=0$

$a_3-a_0=0$

$\frac{a_4}{2!}-a_1=0$

$\frac{a_5}{3!}-\frac{a_2}{2!}=0$

$[n>2]$ $\frac{a_n}{(n-2)!}-\frac{a_{n-3}}{(n-3)!}=0$

$a_n=(n-2)a_{n-3}$

$a_0=c_1$ $a_1=c_2$ $a_2=0$

$a_3=a_0=c_1$

$a_4=2a_1=2c_2$

$a_5=3a_2=0$

if $n>=0$ then $a_{3n+2}=0$

$w(x)=c_1+c_2x+\frac{c_1x^{3}}{3!}+\frac{2c_2x^{4}}{4!}+\frac{4c_1x^{6}}{6!}+\frac{2.5c_2x^{7}}{7!}+\frac{4.7c_1x^{9}}{9!}+\frac{2.5.8c_2x^{10}}{10!}+......$

$w(x)=c_1(1+\frac{x^{3}}{3!}+\frac{4x^{6}}{6!}+\frac{4.7x^{9}}{9!}+.....)+c_2(x+\frac{2x^{4}}{4!}+\frac{2.5x^{7}}{7!}+\frac{2.5.8x^{10}}{10!}+.....)$

$w(x)=c_1(1+\sum_{k=1}^\infty \frac{1.4.7...(3k-2)x^{3k}}{(3k)!})+c_2(x+\sum_{k=1}^\infty \frac{2.5.8...(3k-1)x^{3k+1}}{(3k+1)!})$

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Let's say that $w(z)$ is one such solution and suppose that it has a power series, convergent in some disk around the origin, given by $$ w(z) = \sum_n a_n z^n. $$ You just substitute that into the equation, and get a recurrence relation for the coefficients $a_0,a_1,a_2,\ldots$. But, in fact, you will get a linear recurrence relation that relates coefficients $a_2,a_3,\ldots$ to $a_0$ and $a_1$, so what you get is $$ w(z) = a_0 \sum_n a^{(1)}_n z^n + a_1 \sum_n a^{(2)}_n z^n, $$ where $a^{(1)}$ and $a^{(2)}$ are two distinct linearly independent solutions to the recurrence relation (for some equations it is a little trickier than this). The two power series define the two linearly independent solutions of the ODE.

Finally, a power series is analytic inside its radius of convergence. See here, for example.

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1  
You seem to be going at it in reverse. If it is assumed to be differentiable (which it is!), then you can assume it is analytic, and therefore assume a power series representation. –  Aryabhata Feb 16 '12 at 7:20

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