90.31.23 problem 26

Internal problem ID [25473]
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 : 26
Date solved : Friday, October 03, 2025 at 12:01:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} \left (t +1\right ) y^{\prime \prime }-t \left (2 t +1\right ) y^{\prime }+\left (2 t +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 28

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

\[ y = \left (\left (c_2 \ln \left (t \right )+c_1 \right ) \left (1+\operatorname {O}\left (t^{6}\right )\right )+\left (t +\operatorname {O}\left (t^{6}\right )\right ) c_2 \right ) t \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 19

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

\[ y(t)\to c_2 \left (t^2+t \log (t)\right )+c_1 t \]

✓ Sympy. Time used: 0.379 (sec). Leaf size: 10

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

\[ y{\left (t \right )} = C_{2} t + C_{1} + O\left (t^{6}\right ) \]