59.1.346 problem 353

Internal problem ID [9518]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 353
Date solved : Sunday, March 30, 2025 at 02:36:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 56

ode:=4*(t^2-3*t+2)*diff(diff(y(t),t),t)-2*diff(y(t),t)+y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.176 (sec). Leaf size: 112

ode=4*(t^2-3*t+2)*D[y[t],{t,2}]-2*D[y[t],t]+y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \exp \left (\int _1^t\frac {2 K[1]-5}{4 \left (K[1]^2-3 K[1]+2\right )}dK[1]-\frac {1}{2} \int _1^t-\frac {1}{2 \left (K[2]^2-3 K[2]+2\right )}dK[2]\right ) \left (c_2 \int _1^t\exp \left (-2 \int _1^{K[3]}\frac {2 K[1]-5}{4 \left (K[1]^2-3 K[1]+2\right )}dK[1]\right )dK[3]+c_1\right ) \]

✓ Sympy. Time used: 0.471 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((4*t**2 - 12*t + 8)*Derivative(y(t), (t, 2)) + y(t) - 2*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \sqrt {t - 2} \left (C_{1} \sqrt {\frac {t - 1}{t - 2}} {{}_{1}F_{0}\left (\begin {matrix} 0 \\ \end {matrix}\middle | {\frac {t - 1}{t - 2}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 1 \\ \frac {1}{2} \end {matrix}\middle | {\frac {t - 1}{t - 2}} \right )}\right ) \]