90.31.2 problem 2

Internal problem ID [25452]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 7. Power series methods. Exercises at page 537
Problem number : 2
Date solved : Friday, October 03, 2025 at 12:01:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {\left (1-t \right ) y^{\prime }}{t}+\frac {\left (1-\cos \left (t \right )\right ) y}{t^{3}}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 44

Order:=6; 
ode:=diff(diff(y(t),t),t)+(1-t)/t*diff(y(t),t)+(-cos(t)+1)/t^3*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1-\frac {1}{2} t -\frac {1}{16} t^{2}-\frac {5}{864} t^{3}-\frac {61}{27648} t^{4}-\frac {3239}{6912000} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (2 t +\frac {3}{16} t^{2}+\frac {73}{2592} t^{3}+\frac {1717}{165888} t^{4}+\frac {385843}{207360000} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 115

ode=D[y[t],{t,2}]+(1-t)/t*D[y[t],t]+(1-Cos[t])/t^3*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \left (-\frac {3239 t^5}{6912000}-\frac {61 t^4}{27648}-\frac {5 t^3}{864}-\frac {t^2}{16}-\frac {t}{2}+1\right )+c_2 \left (\frac {385843 t^5}{207360000}+\frac {1717 t^4}{165888}+\frac {73 t^3}{2592}+\frac {3 t^2}{16}+\left (-\frac {3239 t^5}{6912000}-\frac {61 t^4}{27648}-\frac {5 t^3}{864}-\frac {t^2}{16}-\frac {t}{2}+1\right ) \log (t)+2 t\right ) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Derivative(y(t), (t, 2)) + (1 - t)*Derivative(y(t), t)/t + (1 - cos(t))*y(t)/t**3,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE Derivative(y(t), (t, 2)) + (1 - t)*Derivative(y(t), t)/t + (1 - cos(t))*y(t)/t**3 d