65.12.2 problem 19.1 (ii)

Internal problem ID [13718]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 19, CauchyEuler equations. Exercises page 174
Problem number : 19.1 (ii)
Date solved : Wednesday, March 05, 2025 at 10:13:43 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x^{2} y^{\prime \prime }+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: 0.017 (sec). Leaf size: 14

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

\[ y = \sqrt {x}\, \left (1-\frac {\ln \left (x \right )}{2}\right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 47

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

\[ y(x)\to -\frac {1}{3} \sqrt {x} \left (\sqrt {3} \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )-3 \cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )\right ) \]

✓ Sympy. Time used: 0.071 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \sqrt {x} \left (1 - \frac {\log {\left (x \right )}}{2}\right ) \]