59.1.623 problem 640

Internal problem ID [9795]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 640
Date solved : Sunday, March 30, 2025 at 02:46:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

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

\[ y = \sqrt {t}\, \left (c_1 \sin \left (\sqrt {t}\, \sqrt {2}\right )+c_2 \cos \left (\sqrt {t}\, \sqrt {2}\right )\right ) \]

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 62

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

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

✓ Sympy. Time used: 0.223 (sec). Leaf size: 37

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

\[ y{\left (t \right )} = t^{\frac {3}{4}} \left (C_{1} J_{\frac {1}{2}}\left (\sqrt {2} \sqrt {t}\right ) + C_{2} Y_{\frac {1}{2}}\left (\sqrt {2} \sqrt {t}\right )\right ) \]