59.1.409 problem 421

Internal problem ID [9581]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 421
Date solved : Wednesday, March 05, 2025 at 07:51:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 34

ode:=t*diff(diff(y(t),t),t)+(t^2-1)*diff(y(t),t)+t^3*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {t^{2}}{4}} \left (c_{1} \cos \left (\frac {\sqrt {3}\, t^{2}}{4}\right )+c_{2} \sin \left (\frac {\sqrt {3}\, t^{2}}{4}\right )\right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 48

ode=t*D[y[t],{t,2}]+(t^2-1)*D[y[t],t]+t^3*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-\frac {t^2}{4}} \left (c_2 \cos \left (\frac {\sqrt {3} t^2}{4}\right )+c_1 \sin \left (\frac {\sqrt {3} t^2}{4}\right )\right ) \]

✓ Sympy. Time used: 0.788 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**3*y(t) + t*Derivative(y(t), (t, 2)) + (t**2 - 1)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} \left (1 - \frac {t^{4}}{8}\right ) + C_{1} t^{2} \left (1 - \frac {t^{2}}{4}\right ) + O\left (t^{6}\right ) \]