14.13.10 problem 10

Internal problem ID [2637]
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 : 10
Date solved : Sunday, March 30, 2025 at 12:12:14 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

With initial conditions

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 14

ode:=t^2*diff(diff(y(t),t),t)-3*t*diff(y(t),t)+4*y(t) = 0; 
ic:=y(1) = 1, D(y)(1) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 15

ode=t^2*D[y[t],{t,2}]-3*t*D[y[t],t]+4*y[t]==0; 
ic={y[1]==1,Derivative[1][y][1] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to t^2 (1-2 \log (t)) \]

✓ Sympy. Time used: 0.173 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) - 3*t*Derivative(y(t), t) + 4*y(t),0) 
ics = {y(1): 1, Subs(Derivative(y(t), t), t, 1): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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