14.9.11 problem 14

Internal problem ID [2577]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.2. Equal roots, reduction of order. Excercises page 149
Problem number : 14
Date solved : Tuesday, March 04, 2025 at 02:28:22 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=3 t^{2}-1 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 44

ode:=(-t^2+1)*diff(diff(y(t),t),t)-2*t*diff(y(t),t)+6*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 55

ode=(1-t^2)*D[y[t],{t,2}]-2*t*D[y[t],t]+6*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{2} c_1 \left (3 t^2-1\right )-\frac {1}{4} c_2 \left (\left (3 t^2-1\right ) \log (1-t)+\left (1-3 t^2\right ) \log (t+1)+6 t\right ) \]

✓ Sympy. Time used: 0.925 (sec). Leaf size: 26

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)) + 6*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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