Method of Frobenius with exponential coefficients

[phil ess]

I know how to do Frobenius on variable coefficient ODE's but only when the coefficients are powers of the independent variable. Can I do method of Frobenius on something like:

y'' + e^-xy = 0 ?

What form would I assume a solution of? Just the regular y=sum(A_kx^k+r ?

Thanks for the help!

 

[LCKurtz]

I think you would let y = ∑a_nxⁿ, expand the exponential in a series about 0 and use the Cauchy product for the e^-xy term. You should be able to calculate the first few terms by hand and with luck maybe find a pattern.

If it is that specific equation you are interested in, the solution is known. See

http://eqworld.ipmnet.ru/en/solutions/ode/ode0227.pdf