59.1.630 problem 647

Internal problem ID [9802]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 647
Date solved : Wednesday, March 05, 2025 at 07:59:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }+t y^{\prime }+2 y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.203 (sec). Leaf size: 43

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

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

✓ Sympy. Time used: 0.692 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = C_{1} t \left (\frac {t^{4}}{8} - \frac {5 t^{3}}{12} + t^{2} - \frac {3 t}{2} + 1\right ) + O\left (t^{6}\right ) \]