18.3.12 problem Problem 16.13

Internal problem ID [3512]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 16, Series solutions of ODEs. Section 16.6 Exercises, page 550
Problem number : Problem 16.13
Date solved : Sunday, March 30, 2025 at 01:45:26 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+\frac {y}{z^{3}}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✗ Maple

Order:=6; 
ode:=diff(diff(y(z),z),z)+1/z^3*y(z) = 0; 
dsolve(ode,y(z),type='series',z=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 222

ode=D[y[z],{z,2}]+1/z^3*y[z]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[z],{z,0,5}]
 

\[ y(z)\to c_1 e^{-\frac {2 i}{\sqrt {z}}} z^{3/4} \left (-\frac {468131288625 i z^{9/2}}{8796093022208}+\frac {66891825 i z^{7/2}}{4294967296}-\frac {72765 i z^{5/2}}{8388608}+\frac {105 i z^{3/2}}{8192}+\frac {33424574007825 z^5}{281474976710656}-\frac {14783093325 z^4}{549755813888}+\frac {2837835 z^3}{268435456}-\frac {4725 z^2}{524288}+\frac {15 z}{512}-\frac {3 i \sqrt {z}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {z}}} z^{3/4} \left (\frac {468131288625 i z^{9/2}}{8796093022208}-\frac {66891825 i z^{7/2}}{4294967296}+\frac {72765 i z^{5/2}}{8388608}-\frac {105 i z^{3/2}}{8192}+\frac {33424574007825 z^5}{281474976710656}-\frac {14783093325 z^4}{549755813888}+\frac {2837835 z^3}{268435456}-\frac {4725 z^2}{524288}+\frac {15 z}{512}+\frac {3 i \sqrt {z}}{16}+1\right ) \]

✗ Sympy

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq(Derivative(y(z), (z, 2)) + y(z)/z**3,0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE Derivative(y(z), (z, 2)) + y(z)/z**3 does not match hint 2nd_power_series_regular