# Can the 2nd ode x*y''-c*y=0 be solved exactly?

1. Jul 2, 2010

### sufive

As the tittle, can the 2nd ode (B.C not fixed, I need the general solution)

y''-c/x*y=0

be solved exactly? Or can it be translated into some
special mathphysical equations, such as Bessel?
Hypergeometric? etc

any comments or references are welcome.

Thank you for advance!

2. Jul 3, 2010

Would this work?

$$y'' - \frac{cy}{x} = 0$$, can be rewritten as
$$\frac{dydy}{y} = \frac{c}{x}dxdx$$, which upon integrating gives
$$ln|y|dy = c*ln|x|dx$$, and then we would just have to integrate one more time?

3. Jul 3, 2010

### Staff: Mentor

I would go with a series solution, with y = c0 + c1x + c2x2 + ... + cnxn + ...

4. Jul 3, 2010

### gomunkul51

Don't you need to search for a solution not from the type:

$$y(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$$

but from the form:

$$y(x)=\sum_{n=0}^{\infty}a_{n}x^{n+r}$$

$$a_{0}\neq 0$$

because of the singularity at x=0 ?

I don't think it is possible to find both linearly independent series' if you search the solution in the first form, I couldn't. but may be I'm rusty in the Olde ODE's :)

5. Jul 3, 2010

### HallsofIvy

Staff Emeritus
No, that is not valid.
$$\frac{d^2y}{dx^2}\ne \frac{(dy)(dy)}{(dx)(dx)}$$

6. Jul 3, 2010

7. Jul 3, 2010

### Staff: Mentor

The equation as originally given is xy'' - cy = 0, which is defined when x = 0. It is only by dividing by x that you introduce an singularity.

8. Jul 3, 2010

### sufive

Thans to all of you. Your browse or reply helps me.

To this point, LCKurtz's message is the most nearest to
my need!

9. Jul 4, 2010

### LCKurtz

No, this DE has a regular singular point at x = 0. gomunkul51 is correct about the form the infinite series would take. See

http://banach.millersville.edu/~bob/math365/singular.pdf

for the definition of a singular point and discussion.

10. Jul 4, 2010

### vela

Staff Emeritus
You might also try a change of variables to transform the differential equation into the Bessel differential equation. For instance, u=sqrt(x) gets you a differential equation that sort of looks like the Bessel differential equation.

11. Jul 4, 2010