90.31.4 problem 4

Internal problem ID [25454]
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 : 4
Date solved : Friday, October 03, 2025 at 12:01:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 222

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

\[ y(t)\to c_1 e^{-\frac {2 i}{\sqrt {t}}} \sqrt [4]{t} \left (\frac {655383804075 i t^{9/2}}{8796093022208}-\frac {103378275 i t^{7/2}}{4294967296}+\frac {135135 i t^{5/2}}{8388608}-\frac {315 i t^{3/2}}{8192}-\frac {45221482481175 t^5}{281474976710656}+\frac {21606059475 t^4}{549755813888}-\frac {4729725 t^3}{268435456}+\frac {10395 t^2}{524288}-\frac {105 t}{512}-\frac {15 i \sqrt {t}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {t}}} \sqrt [4]{t} \left (-\frac {655383804075 i t^{9/2}}{8796093022208}+\frac {103378275 i t^{7/2}}{4294967296}-\frac {135135 i t^{5/2}}{8388608}+\frac {315 i t^{3/2}}{8192}-\frac {45221482481175 t^5}{281474976710656}+\frac {21606059475 t^4}{549755813888}-\frac {4729725 t^3}{268435456}+\frac {10395 t^2}{524288}-\frac {105 t}{512}+\frac {15 i \sqrt {t}}{16}+1\right ) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Derivative(y(t), (t, 2)) + Derivative(y(t), t)/t + (1 - 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)) + Derivative(y(t), t)/t + (1 - t)*y(t)/t**3 does not match