- #1
arroy_0205
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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:
[tex]
e^{-2y}\left(z''(y)-2z'(y)+\frac{3}{4}z(y)\right)+z(y)=0
[/tex]
We change variable to
[tex]
u=e^y
[/tex]
then the ODE is
[tex]
u^2z''(u)-uz'(u)+z(u)(u^2+\frac{3}{4})=0
[/tex]
The author writes the solution is:
[tex]
z(y)=\frac{1}{u^{3/2}}z(u)=u\left(J_{\frac{1}{2}}(u)+c_1N_{\frac{1}{2}}(u) \right)
[/tex]
where J and N are Bessel functions. My problem is, I am getting the relation
[tex]
z(u)=u\left(c_1 J_{\frac{1}{2}}(u)+c_2 N_{\frac{1}{2}}(u) \right)
[/tex]
But note that according to the author,
[tex]
z(y)={u^{5/2}}\left(J_{\frac{1}{2}}(u)+c_1N_{\frac{1}{2}}(u) \right)
[/tex]
Or in other words though I can get the desired solution in terms of changed variable,
I am not able to get the relation:
[tex]
z(y)=\frac{1}{u^{3/2}}z(u)
[/tex]
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?
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:
[tex]
e^{-2y}\left(z''(y)-2z'(y)+\frac{3}{4}z(y)\right)+z(y)=0
[/tex]
We change variable to
[tex]
u=e^y
[/tex]
then the ODE is
[tex]
u^2z''(u)-uz'(u)+z(u)(u^2+\frac{3}{4})=0
[/tex]
The author writes the solution is:
[tex]
z(y)=\frac{1}{u^{3/2}}z(u)=u\left(J_{\frac{1}{2}}(u)+c_1N_{\frac{1}{2}}(u) \right)
[/tex]
where J and N are Bessel functions. My problem is, I am getting the relation
[tex]
z(u)=u\left(c_1 J_{\frac{1}{2}}(u)+c_2 N_{\frac{1}{2}}(u) \right)
[/tex]
But note that according to the author,
[tex]
z(y)={u^{5/2}}\left(J_{\frac{1}{2}}(u)+c_1N_{\frac{1}{2}}(u) \right)
[/tex]
Or in other words though I can get the desired solution in terms of changed variable,
I am not able to get the relation:
[tex]
z(y)=\frac{1}{u^{3/2}}z(u)
[/tex]
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: