14.13.3 problem 3

Internal problem ID [2630]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.1, singular points, Euler equations. Excercises page 203
Problem number : 3
Date solved : Tuesday, March 04, 2025 at 02:32:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 15

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

\[ y = \left (-1+t \right ) \left (\left (-1+t \right ) c_1 +c_2 \right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 18

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

\[ y(t)\to (t-1) (c_2 (t-1)+c_1) \]

✓ Sympy. Time used: 0.249 (sec). Leaf size: 10

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

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