14.14.5 problem 5

Internal problem ID [2642]
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 : 5
Date solved : Sunday, March 30, 2025 at 12:12:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} -1 \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.087 (sec). Leaf size: 111

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

\[ y(t)\to c_2 e^{\frac {1}{2 (t+1)}} \left (\frac {516353141702117 (t+1)^5}{33443020800}+\frac {53349163853 (t+1)^4}{39813120}+\frac {58276991 (t+1)^3}{414720}+\frac {21397 (t+1)^2}{1152}+\frac {79 (t+1)}{24}+1\right ) (t+1)^{7/4}+c_1 \left (\frac {53}{5} (t+1)^5-\frac {25}{12} (t+1)^4+\frac {2}{3} (t+1)^3-\frac {1}{2} (t+1)^2+1\right ) \]

✗ Sympy

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

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