Solving an ODE with variable coefficients

[c0der]

Homework Statement

Solve the following:
[/B]
y'' = c₂ / (x² + c₁*x) * y
c₁, c₂ are constants, x is variable

Homework Equations

As above

The Attempt at a Solution

I have used the method of Frobenius and regular power series and obtained an infinite series on top of an infinite series, which is indefinite. What are the normal methods to solve that ODE? It's not quite in the Frobenius or Bessel forms

 

[ShayanJ]

I used a regular power series and there was no problem. Just multiply the equation by $x^2+c_1 x$ to get $x^2 y''+c_1 x y''-c_2 y=0$.
The regular series gives only one of the solutions.So you should also consider a solution of the form $y=\sum_{n=0}^\infty a_n x^{n+m}$.

 

[Ray Vickson]

Homework Statement

Solve the following:
[/B]
y'' = c₂ / (x² + c₁*x) * y
c₁, c₂ are constants, x is variable

Homework Equations

As above

The Attempt at a Solution

I have used the method of Frobenius and regular power series and obtained an infinite series on top of an infinite series, which is indefinite. What are the normal methods to solve that ODE? It's not quite in the Frobenius or Bessel forms

Maple gets a solution in terms of hypergeometric functions with (variable) argument $-c_1/x$. Thus, the power-series expansion of Maple's solution will be in powers of $1/x$, not of $x$ itself.

 

[c0der]

Thank you, I have done this however as follows:

Equating coefficients of x⁰ gives:
a₀=0

For x¹:

a₂ = c₂/2c₁

Equating coefficients of x^m:

a_m+1 = [ c₂ - (m-1)m ] / [ c₁m(m+1) ] a_m for m>=2

Then:

a₃ = [ (c₂ - 2) / 3!2!c₁² ] a₁

a₄ = [ (c₂ - 6)(c₂ - 2)c₂ / 4!3!c₁³ ] a₁

a₅ = [ (c₂ - 12)(c₂ - 6)(c₂ - 2)c₂ / 5!4!c₁⁴ ] a₁

a₆ = [ (c₂ - 20)(c₂ - 12)(c₂ - 6)(c₂ - 2)c₂ / 6!5!c₁⁵ ] a₁

How can I correlate this with the hypergeometric series or any other recognizable series? The numerator terms in the brackets have a pattern to it 2+4=6 6+6=12 12+8=20 etc but getting the terms in powers of m is difficult

 

[haruspex]

y'' = c₂ / (x² + c₁*x) * y

Is that (c₂ / (x² + c₁*x)) * y or c₂ / ((x² + c₁*x) * y)?

 

[c0der]

(c₂ / (x² + c₁x)) * y