90.31.1 problem 1

Internal problem ID [25451]
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 : 1
Date solved : Friday, October 03, 2025 at 12:01:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {t y^{\prime }}{-t^{2}+1}+\frac {y}{t +1}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 54

Order:=6; 
ode:=diff(diff(y(t),t),t)+t/(-t^2+1)*diff(y(t),t)+1/(t+1)*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \left (1-\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}-\frac {1}{120} t^{5}\right ) y \left (0\right )+\left (t -\frac {1}{3} t^{3}+\frac {1}{12} t^{4}-\frac {1}{30} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(t)\to c_2 \left (-\frac {t^5}{30}+\frac {t^4}{12}-\frac {t^3}{3}+t\right )+c_1 \left (-\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}-\frac {t^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.415 (sec). Leaf size: 60

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

\[ y{\left (t \right )} = - \frac {t^{4} r{\left (3 \right )}}{2} + \frac {2 t^{5} r{\left (3 \right )}}{5} + C_{2} \left (- \frac {3 t^{5}}{40} + \frac {t^{4}}{8} - \frac {t^{2}}{2} + 1\right ) + C_{1} t \left (\frac {t^{4}}{10} - \frac {t^{3}}{12} + 1\right ) + O\left (t^{6}\right ) \]