13.9.9 problem 12

Internal problem ID [2395]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 12
Date solved : Sunday, March 30, 2025 at 12:00:14 AM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 33

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

\[ y(t)\to c_1 t-\frac {1}{2} c_2 (t \log (1-t)-t \log (t+1)+2) \]

✓ Sympy. Time used: 0.964 (sec). Leaf size: 20

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

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