7740.10.2
Frobenius method for boundary-value problems

a) Suggested transformation and differential equation in v[x]

Solve DE using suggested transformation (solution is polynomial) => confluent hypergeometric functions of the first kind

Limits - boundary behavior

Finite y[0] => C[1]=0

Limit at origin => regular singular point

d) Attempt to evaluate limit at infinity (note that decreases faster than any polynomial in x) [note the Bessel case for e = 0 and hypergeometric case for e = 2]

Empirical search for stable eigensolutions (that approach zero as x approaches infinity)

Negative e

Zero e

Positive e

Fixed negative C[2]

Fixed positive C[2]

Systematic search for eigensolutions

Construct the characteristic equation and solve for eigenvalues

Series[y[x]/.soln/.C[1]->0, {x, 0, 7}]