59.1.206 problem 209

Internal problem ID [9378]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 209
Date solved : Wednesday, March 05, 2025 at 07:48:19 AM
CAS classification : [_Lienard]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.15 (sec). Leaf size: 56

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

\[ y(t)\to \frac {1}{2} e \left (\sqrt {2 \pi } c_2 e^{\frac {t^2}{2}} \text {erf}\left (\frac {t}{\sqrt {2}}\right )+2 c_1 e^{\frac {t^2}{2}}-2 c_2 t\right ) \]

✓ Sympy. Time used: 0.815 (sec). Leaf size: 31

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

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