14.14.6 problem 6

Internal problem ID [2643]
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 : 6
Date solved : Sunday, March 30, 2025 at 12:12:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{3} y^{\prime \prime }+\sin \left (t^{2}\right ) y^{\prime }+t y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.144 (sec). Leaf size: 31

Order:=6; 
ode:=t^3*diff(diff(y(t),t),t)+sin(t^2)*diff(y(t),t)+t*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = c_1 \,t^{-i} \left (1+\left (\frac {1}{240}-\frac {i}{120}\right ) t^{4}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \,t^{i} \left (1+\left (\frac {1}{240}+\frac {i}{120}\right ) t^{4}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 68

ode=t^3*D[y[t],{t,2}]+Sin[t^2]*D[y[t],t]+t*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 t^{-i} \left (\left (\frac {1}{23040}-\frac {i}{11520}\right ) t^8+\left (\frac {1}{240}-\frac {i}{120}\right ) t^4+1\right )+c_2 t^i \left (\left (\frac {1}{23040}+\frac {i}{11520}\right ) t^8+\left (\frac {1}{240}+\frac {i}{120}\right ) t^4+1\right ) \]

✗ Sympy

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

ValueError : ODE t**3*Derivative(y(t), (t, 2)) + t*y(t) + sin(t**2)*Derivative(y(t), t) does not match hint 2nd_power_series_regular