59.1.624 problem 641

Internal problem ID [9796]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 641
Date solved : Sunday, March 30, 2025 at 02:46:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 47

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

\[ y = \frac {{\mathrm e}^{-\frac {t}{2}} \left (\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {-t}}{2}\right ) \sqrt {\pi }\, 2^{{3}/{4}} c_1 t +4 c_2 \sqrt {t}\, \sqrt {-t}\right )}{4 \sqrt {-t}} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 46

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

\[ y(t)\to e^{-t/2} \left (c_2 \sqrt {t}+\sqrt {2} c_1 \sqrt {-t} \Gamma \left (\frac {1}{2},-\frac {t}{2}\right )\right ) \]

✓ Sympy. Time used: 1.485 (sec). Leaf size: 474

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

\[ \text {Solution too large to show} \]