59.1.12 problem 12

Internal problem ID [9184]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 12
Date solved : Sunday, March 30, 2025 at 02:25:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 82

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

\[ y = -\frac {\sqrt {2}\, {\mathrm e}^{-\frac {t \left (t -2\right )}{2}} \left (-c_2 \sqrt {\pi }\, \left (-1+t \right ) \left (t -2\right ) \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {-\left (t -2\right )^{2}}}{2}\right )+\sqrt {2}\, \left (c_2 \,{\mathrm e}^{\frac {\left (t -2\right )^{2}}{2}}-t c_1 +c_1 \right ) \sqrt {-\left (t -2\right )^{2}}\right )}{2 \sqrt {-\left (t -2\right )^{2}}} \]

✓ Mathematica. Time used: 0.557 (sec). Leaf size: 54

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

\[ y(t)\to e^{t-\frac {t^2}{2}} (t-1) \left (c_2 \int _1^t\frac {e^{\frac {1}{2} (K[1]-4) K[1]} K[1]}{(K[1]-1)^2}dK[1]+c_1\right ) \]

✓ Sympy. Time used: 0.826 (sec). Leaf size: 36

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{5}}{15} - \frac {t^{3}}{3} + 1\right ) + C_{1} t^{2} \left (- \frac {t^{3}}{15} - \frac {t^{2}}{4} + 1\right ) + O\left (t^{6}\right ) \]