14.14.27 problem 27

Internal problem ID [2664]
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 : 27
Date solved : Sunday, March 30, 2025 at 12:13:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.134 (sec). Leaf size: 44

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

\[ y = c_1 \sqrt {t}\, \left (1+\frac {5}{6} t +\frac {17}{60} t^{2}+\frac {89}{1260} t^{3}+\frac {941}{45360} t^{4}+\frac {14989}{2494800} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+2 t +t^{2}+\frac {4}{15} t^{3}+\frac {1}{14} t^{4}+\frac {101}{4725} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 79

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

\[ y(t)\to c_1 \sqrt {t} \left (\frac {14989 t^5}{2494800}+\frac {941 t^4}{45360}+\frac {89 t^3}{1260}+\frac {17 t^2}{60}+\frac {5 t}{6}+1\right )+c_2 \left (\frac {101 t^5}{4725}+\frac {t^4}{14}+\frac {4 t^3}{15}+t^2+2 t+1\right ) \]

✗ Sympy

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

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