65.12.2 problem 19.1 (ii)

Internal problem ID [13797]
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 : Tuesday, January 28, 2025 at 06:03:47 AM
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*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 14

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

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

✓ Solution by Mathematica

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

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

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