14.14.4 problem 4

Internal problem ID [2641]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.2, Regular singular points, the method of Frobenius. Excercises page 216
Problem number : 4
Date solved : Sunday, March 30, 2025 at 12:12:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.065 (sec). Leaf size: 44

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

\[ y = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1-t +\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{24} t^{4}-\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (\frac {3}{2} t -\frac {23}{24} t^{2}+\frac {3}{8} t^{3}-\frac {301}{2880} t^{4}+\frac {13}{576} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 113

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

\[ y(t)\to c_1 \left (-\frac {t^5}{120}+\frac {t^4}{24}-\frac {t^3}{6}+\frac {t^2}{2}-t+1\right )+c_2 \left (\frac {13 t^5}{576}-\frac {301 t^4}{2880}+\frac {3 t^3}{8}-\frac {23 t^2}{24}+\left (-\frac {t^5}{120}+\frac {t^4}{24}-\frac {t^3}{6}+\frac {t^2}{2}-t+1\right ) \log (t)+\frac {3 t}{2}\right ) \]

✗ Sympy

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

ValueError : ODE (exp(t) - 1)*Derivative(y(t), (t, 2)) + y(t) + exp(t)*Derivative(y(t), t) does not match hint 2nd_power_series_regular