90.31.20 problem 23

Internal problem ID [25470]
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 : 23
Date solved : Friday, October 03, 2025 at 12:01:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 52

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

\[ y(t)\to \left (\frac {1}{5}+\frac {2 i}{5}\right ) c_2 t^{-i} ((1+2 i) t+(1-2 i))-\left (\frac {2}{5}+\frac {i}{5}\right ) c_1 t^i ((2+i) t-(2-i)) \]

✓ Sympy. Time used: 0.406 (sec). Leaf size: 10

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

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