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Doubts about change of variable in ODE

  1. Jul 13, 2008 #1
    I was trying to check some calculations from a paper myself and I got
    stuck at a differential equation. Can anybody help me with suggestions?
    I am giving a simpler version of the differential eqn by setting all numerical
    constants=1. Consider:
    We change variable to
    then the ODE is
    The author writes the solution is:
    z(y)=\frac{1}{u^{3/2}}z(u)=u\left(J_{\frac{1}{2}}(u)+c_1N_{\frac{1}{2}}(u) \right)
    where J and N are Bessel functions. My problem is, I am getting the relation
    z(u)=u\left(c_1 J_{\frac{1}{2}}(u)+c_2 N_{\frac{1}{2}}(u) \right)
    But note that according to the author,
    z(y)={u^{5/2}}\left(J_{\frac{1}{2}}(u)+c_1N_{\frac{1}{2}}(u) \right)
    Or in other words though I can get the desired solution in terms of changed variable,
    I am not able to get the relation:
    at all. How to get this relation? Also, note that the author uses one constant
    in the solution but I think there should be two independent constants. What am I missing in the calculations?
    Last edited: Jul 13, 2008
  2. jcsd
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