I was trying to check some calculations from a paper myself and I got(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Doubts about change of variable in ODE

Can you offer guidance or do you also need help?

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