59.1.622 problem 639

Internal problem ID [9794]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 639
Date solved : Wednesday, March 05, 2025 at 07:59:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 56

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

\[ y = c_{1} \sqrt {\pi }\, \left (t^{2}+6 t +3\right ) \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {t}}{2}\right )+5 c_{1} \sqrt {2}\, \left (\sqrt {t}+\frac {t^{{3}/{2}}}{5}\right ) {\mathrm e}^{-\frac {t}{2}}+c_{2} \left (t^{2}+6 t +3\right ) \]

✓ Mathematica. Time used: 0.308 (sec). Leaf size: 58

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

\[ y(t)\to \left (t^2+6 t+3\right ) \left (c_2 \int _1^t\frac {e^{-\frac {K[1]}{2}-\frac {1}{2}}}{\sqrt {K[1]} \left (K[1]^2+6 K[1]+3\right )^2}dK[1]+c_1\right ) \]

✓ Sympy. Time used: 0.802 (sec). Leaf size: 44

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{2}}{3} + 2 t + 1\right ) + C_{1} \sqrt {t} \left (\frac {t^{4}}{40320} - \frac {t^{3}}{1680} + \frac {t^{2}}{40} + \frac {t}{2} + 1\right ) + O\left (t^{6}\right ) \]