90.31.3 problem 3

Internal problem ID [25453]
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 : 3
Date solved : Friday, October 03, 2025 at 12:01:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{2} t^{2}+\frac {1}{12} t^{3}-\frac {7}{36} t^{4}+\frac {21}{160} t^{5}\right ) y \left (0\right )+\left (t -\frac {1}{3} t^{3}+\frac {7}{24} t^{4}+\frac {17}{120} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

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

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

\[ y(t)\to c_2 \left (\frac {17 t^5}{120}+\frac {7 t^4}{24}-\frac {t^3}{3}+t\right )+c_1 \left (\frac {21 t^5}{160}-\frac {7 t^4}{36}+\frac {t^3}{12}+\frac {t^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.448 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*t*(1 - t)*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + (1 - exp(t))*y(t)/t,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} t \left (\frac {9 t^{4}}{40} + \frac {t^{3}}{4} - \frac {t^{2}}{2} + 1\right ) + C_{1} + O\left (t^{6}\right ) \]