59.1.621 problem 638

Internal problem ID [9793]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 638
Date solved : Wednesday, March 05, 2025 at 07:59:04 AM
CAS classification : [_Laguerre]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 15

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

\[ y = {\mathrm e}^{t} \left (\operatorname {erf}\left (\sqrt {t}\right ) c_{1} +c_{2} \right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 21

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

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

✓ Sympy. Time used: 0.911 (sec). Leaf size: 65

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{5}}{120} + \frac {t^{4}}{24} + \frac {t^{3}}{6} + \frac {t^{2}}{2} + t + 1\right ) + C_{1} \sqrt {t} \left (\frac {16 t^{4}}{945} + \frac {8 t^{3}}{105} + \frac {4 t^{2}}{15} + \frac {2 t}{3} + 1\right ) + O\left (t^{6}\right ) \]