90.31.22 problem 25

Internal problem ID [25472]
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 : 25
Date solved : Friday, October 03, 2025 at 12:01:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 46

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

\[ y = \left (1+t +\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}\right ) \left (c_1 \,t^{-i}+c_2 \,t^{i}\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 62

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

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

✓ Sympy. Time used: 0.365 (sec). Leaf size: 10

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

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