Malmstén's Differential Equation

Author

Eric W. Weisstein
February 27, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/MalmstensDifferentialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MalmstensDifferentialEquation.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Solution

eqn=y''[z]+(r/z)y'[z]-(A (z^m)+(s/(z^2)))y[z];

TraditionalForm[eqn==0]

DSolve[eqn==0,y[z],z]//FullSimplify

y[z_]:=z^((1-r)/2)*BesselI[Sqrt[(-1+r)^2+4*s]/(2+m),(2*Sqrt[A]*z^(1+m/2))/(2+m)]*C[2]+z^((1-r)/2)*BesselI[-(Sqrt[(-1+r)^2+4*s]/(2+m)),(2*Sqrt[A]*z^(1+m/2))/(2+m)]*C[1]

y[z]//TraditionalForm

Chop[Collect[y''[z]+(r/z)y'[z]-(A (z^m)+(s/(z^2)))y[z]/.{s->Random[Complex]}/.{A->Random[Complex]}/.{m->Random[Complex]}/.{r->Random[Complex]}/.{z->Random[Complex]},C[1]]]