How to Solve a 2nd Order Differential Equation with Singular Points?

[jhon]

I can not how define way to solve these equation

x(l-x)y''+4y'+2y=o

If anyone can help

 

[HallsofIvy]

That looks like a pretty standard kind of equation. If you are given an initial value problem with y(a), y'(a) given and a is neither 0 nor 1, a standard series solution will work:
$$y= \sum_{n=0}^\infty a_nx^n$$.

Differentiate term by term and put into the equation to get a recursive equation for $a_n$.

If you are given y(0) and y'(0) you will need to use Frobenius' method with
$$y= \sum_{n=0}^\infty a_n x^{n+ c}$$
for some number c (not necessarily a positive integer). Put that into the d.e. and look at the n=0 term to determine c.

If you are given y(l) and y'(l), similarly you will need to use Frobenius' method with
[tex]y= \sum_{n=0}^\infty a_n (x- l)^{n+ c}[/itex][/tex]

 

[jhon]

thnks HallsofIvy

but the question doesn't give me the initial value

 

[frggr]

I could be wrong (Still just a student) but even w/o initial conditions the series method should work. Just you won't know a₀ and a₁

 

[HallsofIvy]

Yes, but x= 0 and x= l are "singular" points. You may find that for some $a_0$ and $a_1$ your solution cannot be extended to x= 0 or x= 1.