59.1.15 problem 15

Internal problem ID [9187]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 15
Date solved : Sunday, March 30, 2025 at 02:25:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 12

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

\[ y = c_1 t +c_2 \,{\mathrm e}^{t} \]

✓ Mathematica. Time used: 0.17 (sec). Leaf size: 90

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

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

✓ Sympy. Time used: 0.820 (sec). Leaf size: 27

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{4}}{24} + \frac {t^{3}}{6} + \frac {t^{2}}{2} + 1\right ) + C_{1} t + O\left (t^{6}\right ) \]