# Homework Help: 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.

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

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

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