59.1.348 problem 355

Internal problem ID [9520]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 355
Date solved : Sunday, March 30, 2025 at 02:36:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 t \left (1+t \right ) y^{\prime \prime }+t y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 66

ode:=3*t*(t+1)*diff(diff(y(t),t),t)+t*diff(y(t),t)-y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = c_1 t -2 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (1+t \right )^{{1}/{3}}\right ) \sqrt {3}}{3}\right ) t c_2 -6 \left (1+t \right )^{{2}/{3}} c_2 -2 \ln \left (\left (1+t \right )^{{1}/{3}}-1\right ) t c_2 +\ln \left (\left (1+t \right )^{{2}/{3}}+\left (1+t \right )^{{1}/{3}}+1\right ) t c_2 \]

✓ Mathematica. Time used: 0.424 (sec). Leaf size: 78

ode=3*t*(1+t)*D[y[t],{t,2}]+t*D[y[t],t]-y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.792 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*t*(t + 1)*Derivative(y(t), (t, 2)) + t*Derivative(y(t), t) - y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} t \left (\frac {t^{4}}{2880} + \frac {t^{3}}{144} + \frac {t^{2}}{12} + \frac {t}{2} + 1\right ) + O\left (t^{6}\right ) \]