Series solution to DE

**arbolis** · December 15th 2011, 01:30 PM

Consider the DE $xy''+(2-x)y'-2y=0$ . Give 2 solutions; one regular and worth 1 at the origin and the other of the form $\frac{1}{x}+A(x) \ln (x)+B(x)$ where $A(x)$ and $B(x)$ are regular at the origin. Give the first 3 terms of the series of $A(x)$ and $B(x)$ .
My attempt: Divide the DE by x: $y''+ \left ( \frac{2}{x}-1 \right ) y'-2y=0$ . In order to solve this DE, I had in mind to propose a solution of the form $y(x)=f(x) \phi (x)$ where $\phi (x)$ would be the solution to this DE but when x tends to infinity. It turns out that this didn't simplify things as I'd hoped.
When $x\to \infty$ , the DE becomes $\phi ''-\phi '-2\phi =0$ . I used Frobenius's method to solve this DE:
I assumed that $\phi (x)=\sum_{n=0}^{\infty}a_nx^{n+c}$ . I derivated this once and twice and plugged into the DE.
I eventually reached $x^{c-2}a_0c(c-1)+x^{c-1}a_1(c+1)c-x^{c-1}a_0c+\sum_{n=0}^{\infty}x^{n+c}[a_{n+2}(n+2+c)(n+1+c)+a_{n+1}(n+1+c)-2a_n]=0$ .
The indicial equation leads to $c=1$ or $c=0$ . At first glance it looks like both solutions are acceptable.
So now I get a recurrence relation with $a_{n+2}$ in terms of $a_{n+1}$ and $a_n$ which isn't what I hoped for. Maybe I shouldn't have proposed a solution of the form $y(x)=f(x)\phi (x)$ ? How would you tackle this problem?

**arbolis** · December 17th 2011, 09:30 AM

Another attempt using Frobenius method on the original DE.
I reach $x^{c-2}[a_0c(c-1)+2a_0c]+x^{c-1}[a_1c(c+1)+2a_1(c+1)]-a_0x^{c-1}c+\sum_{n=2}^{\infty}[a_n(n+c)(n+c-1)+2a_n(n+c)-a_{n-1}(n+c-1)-2a_{n-2}]=0$ .
Solving the inidicial equation leads me to $c=0$ , or $c=-2$ or $c=-1$ . Though one would expect at most 2 different values for c since it's a second order DE. Something's definitely wrong with my attempt.
Any help is appreciated.

**chisigma** · December 18th 2011, 12:45 AM

Originally Posted by arbolis

Consider the DE $xy''+(2-x)y'-2y=0$ . Give 2 solutions; one regular and worth 1 at the origin and the other of the form $\frac{1}{x}+A(x) \ln (x)+B(x)$ where $A(x)$ and $B(x)$ are regular at the origin. Give the first 3 terms of the series of $A(x)$ and $B(x)$ .
My attempt: Divide the DE by x: $y''+ \left ( \frac{2}{x}-1 \right ) y'-2y=0$ . In order to solve this DE, I had in mind to propose a solution of the form $y(x)=f(x) \phi (x)$ where $\phi (x)$ would be the solution to this DE but when x tends to infinity. It turns out that this didn't simplify things as I'd hoped.
When $x\to \infty$ , the DE becomes $\phi ''-\phi '-2\phi =0$ . I used Frobenius's method to solve this DE:
I assumed that $\phi (x)=\sum_{n=0}^{\infty}a_nx^{n+c}$ . I derivated this once and twice and plugged into the DE.
I eventually reached $x^{c-2}a_0c(c-1)+x^{c-1}a_1(c+1)c-x^{c-1}a_0c+\sum_{n=0}^{\infty}x^{n+c}[a_{n+2}(n+2+c)(n+1+c)+a_{n+1}(n+1+c)-2a_n]=0$ .
The indicial equation leads to $c=1$ or $c=0$ . At first glance it looks like both solutions are acceptable.
So now I get a recurrence relation with $a_{n+2}$ in terms of $a_{n+1}$ and $a_n$ which isn't what I hoped for. Maybe I shouldn't have proposed a solution of the form $y(x)=f(x)\phi (x)$ ? How would you tackle this problem?

A little 'excamotage' that can make the task more comfortable is to set...

$y(x)=\frac{\phi(x)}{x}$ (1)

... and then try to find $\phi(*)$ . First You compute ...

$y^{'}= \frac{\phi^{'}}{x}-\frac{\phi}{x^{2}}$ (2)

$y^{''}=\frac{\phi^{''}}{x}-2\ \frac{\phi^{'}}{x^{2}} +2\ \frac{\phi}{x^{3}}$ (3)

... and then insert (1),(2) and (3) in the original DE obtaining...

$\phi^{''} - \phi^{'} - \frac{\phi}{x} =0$ (4)

... which seems 'easier to be attacked'. Now You search a series solution of (4) imposing proper constrain...

Marry Christmas from Serbia

$\chi$ $\sigma$

**arbolis** · December 18th 2011, 12:46 PM

Thanks for the tips.
I propose a solution of the form $\phi (x)=\sum_{n=0}^{\infty}a_nx^{n+c}$ . After derivating once and twice and plugging into the DE the indicial equation gives me $c=0$ or $c=1$ . And also $a_{n+1}=\frac{a_n}{n+c}$ . But the denominator blows up if $c=0$ and $n=0$ , thus $c=1$ only. Thus $a_{n+1}=\frac{a_n}{n+1}$ .
$a_0$ is arbitrary but different from 0. I'm guessing I have to choose it so that $y (0)=1$ as requested by the statement of the problem. When I choose $a_0=1$ , I get $a_1=1$ , $a_2=\frac{1}{2}$ and in general, $a_n=\frac{1}{n!}$ . So that $\phi (x)=\sum _{n=0}^\infty \frac{x^{n+1}}{n!} \Rightarrow y(x)=\sum _{n=0}^{\infty} \frac{x^n}{n!}=e^x$ . It seems like I got lucky this time when I chose the arbitrary constant $a_0$ since $y(0)=1$ as required.
Is what I've done correct so far?

P.S.:This indeed satisfies the original DE. Wow! Thank you very much!
Post Scriptum 2: I have no idea how to find A(x) and B(x) for a second solution. I tried to write them as infinite series and plug into the DE but I get an enormous expression that doesn't look easily solvable. I'm missing a trick here.

**chisigma** · December 18th 2011, 10:05 PM

Originally Posted by arbolis

Thanks for the tips.
I propose a solution of the form $\phi (x)=\sum_{n=0}^{\infty}a_nx^{n+c}$ . After derivating once and twice and plugging into the DE the indicial equation gives me $c=0$ or $c=1$ . And also $a_{n+1}=\frac{a_n}{n+c}$ . But the denominator blows up if $c=0$ and $n=0$ , thus $c=1$ only. Thus $a_{n+1}=\frac{a_n}{n+1}$ .
$a_0$ is arbitrary but different from 0. I'm guessing I have to choose it so that $y (0)=1$ as requested by the statement of the problem. When I choose $a_0=1$ , I get $a_1=1$ , $a_2=\frac{1}{2}$ and in general, $a_n=\frac{1}{n!}$ . So that $\phi (x)=\sum _{n=0}^\infty \frac{x^{n+1}}{n!} \Rightarrow y(x)=\sum _{n=0}^{\infty} \frac{x^n}{n!}=e^x$ . It seems like I got lucky this time when I chose the arbitrary constant $a_0$ since $y(0)=1$ as required.
Is what I've done correct so far?

P.S.:This indeed satisfies the original DE. Wow! Thank you very much!
Post Scriptum 2: I have no idea how to find A(x) and B(x) for a second solution. I tried to write them as infinite series and plug into the DE but I get an enormous expression that doesn't look easily solvable. I'm missing a trick here.

If a solution of the second order DE...

$\phi^{''} - \phi^{'} - \frac{\phi}{x}=0$ (1)

... is known, a second solution independent from it can be found with the following procedure. If u(x) and v(x) are solutions of (1), then is...

$u^{''} - u^{'} - \frac{u}{x}=0$ (2)

$v^{''} - v^{'} - \frac{v}{x}=0$ (3)

If You multiply the (2) by v, the (3) by u and do the difference You obtain...

$u^{''}\ v -u\ v^{''} + u\ v^{'} - u^{'}\ v = \frac{d}{dx} (u\ v^{'}- u^{'}\ v )=0$ (4)

The (4) is a DE the solution of which is...

$u\ v^{'}- u^{'}\ v=c_{2}$ (5)

... and from (5), deviding by $v^{2}$ You obtain...

$\frac{u\ v^{'}- u^{'}\ v}{v^{2}}= \frac{d}{dx} (\frac{u}{v})= \frac{c_{2}}{v^{2}}$ (6)

Now the solution of (6) is...

$\frac{u}{v}= c_{1}+ c_{2}\ \int \frac{dx}{v^{2}}$ (7)

... and that permits You to derive u(x) from v(x)...

$u= v\ \int \frac{dx}{v^{2}}$ (8)

You found $v(x)=x\ e^{x}$ as possible solution of (1) and from (8) You derive [finally!]...

$y(x)= \frac{u(x)}{x} = e^{x}\ \int \frac{e^{- 2 x}}{x^{2}}\ dx$ (9)

Now all that You have to do is solving the integral in (9)...

Marry Christmas from Serbia

$\chi$ $\sigma$

**chisigma** · December 19th 2011, 01:30 AM

In the previous post we are arrived to write...

$y(x)= e^{x}\ \int \frac{e^{-2 x}}{x^{2}}\ dx$ (1)

The next step is of course the computation of the integral in (1). Proceeding with standard integration by parts we obtain...

$\int \frac{e^{-2 x}}{x^{2}}\ dx= - \frac{e^{-2 x}}{x}- 2\ \int \frac {e^{-2 x}}{x}\ dx = - \frac{e^{-2 x}}{x}- 2\ \text{Ei} (-2 x)$ (2)

... where Ei(*) is the so called Exponential Integral Function, the expansion of which is given by...

$\text{Ei} (x)= \gamma + \ln |x| + \sum_{n=1}^{\infty} \frac{x^{n}}{n\ n!}$ (3)

... where $\gamma$ is the Euler's constant. It is requested to write y(x) in the form...

$y(x)= \frac{1}{x} + \ln |x|\ a(x) + b(x)$ (4)

... and then to find the series expansion of a(x) and b(x). From (1) (2) and (3) we derive [neglecting the sign '-']...

$y(x)= \frac{e^{-x}}{x} +2\ e^{x}\ \text{Ei} (-2 x) =$

$=\frac{e^{-x}}{x} + 2\ e^{x}\ \ln |x| + 2\ e^{x}\ \{\gamma + \ln 2 + \sum_{n=1}^{\infty} \frac{(-2)^{n}\ x^{n}}{n\ n!}\}$ (5)

At this point the effective computation of the first coefficients of a(x) and b(x) is not too complex and the task is left to the Reader...

Marry Christmas from Serbia

$\chi$ $\sigma$

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