59.1.620 problem 636

Internal problem ID [9792]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 636
Date solved : Sunday, March 30, 2025 at 02:46:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4+4 t \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.129 (sec). Leaf size: 60

ode:=diff(diff(y(t),t),t)+(t^2+2*t+1)*diff(y(t),t)-(4+4*t)*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (1+t \right ) \left (t^{3}+3 t^{2}+3 t +5\right ) \left (c_2 \int \frac {{\mathrm e}^{-\frac {t \left (t^{2}+3 t +3\right )}{3}}}{\left (t^{3}+3 t^{2}+3 t +5\right )^{2} \left (1+t \right )^{2}}d t +c_1 \right ) \]

✓ Mathematica. Time used: 0.373 (sec). Leaf size: 78

ode=D[y[t],{t,2}]+(t^2+2*t+1)*D[y[t],t]-(4+4*t)*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to (t+1) \left (t^3+3 t^2+3 t+5\right ) \left (c_2 \int _1^t\frac {e^{-\frac {1}{3} K[1] \left (K[1]^2+3 K[1]+3\right )}}{(K[1]+1)^2 \left (K[1]^3+3 K[1]^2+3 K[1]+5\right )^2}dK[1]+c_1\right ) \]

✓ Sympy. Time used: 0.929 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-4*t - 4)*y(t) + (t**2 + 2*t + 1)*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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