59.1.415 problem 427

Internal problem ID [9587]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 427
Date solved : Sunday, March 30, 2025 at 02:38:07 PM
CAS classification : [_Laguerre]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 13

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

\[ y = c_2 \,{\mathrm e}^{t}+c_1 t +c_1 \]

✓ Mathematica. Time used: 0.406 (sec). Leaf size: 78

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

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

✓ Sympy. Time used: 0.779 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = C_{2} \left (t + 1\right ) + C_{1} t^{2} \left (\frac {t^{3}}{60} + \frac {t^{2}}{12} + \frac {t}{3} + 1\right ) + O\left (t^{6}\right ) \]