65.12.5 problem 19.1 (v)

Internal problem ID [13800]
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 (v)
Date solved : Tuesday, January 28, 2025 at 06:03:54 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} z^{\prime \prime }+3 x z^{\prime }+4 z&=0 \end{align*}

With initial conditions

\begin{align*} z \left (1\right )&=0\\ z^{\prime }\left (1\right )&=5 \end{align*}

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 19

dsolve([x^2*diff(z(x),x$2)+3*x*diff(z(x),x)+4*z(x)=0,z(1) = 0, D(z)(1) = 5],z(x), singsol=all)
 

\[ z = \frac {5 \sqrt {3}\, \sin \left (\sqrt {3}\, \ln \left (x \right )\right )}{3 x} \]

✓ Solution by Mathematica

Time used: 0.090 (sec). Leaf size: 220

DSolve[{x^2*D[z[x],{x,2}]+3*x*z[x]+4*z[x]==0,{z[1]==0,Derivative[1][z][1]==5}},z[x],x,IncludeSingularSolutions -> True]
 

\[ z(x)\to \frac {10 \sqrt {x} \left (\operatorname {BesselJ}\left (i \sqrt {15},2 \sqrt {3}\right ) \operatorname {BesselJ}\left (-i \sqrt {15},2 \sqrt {3} \sqrt {x}\right )-\operatorname {BesselJ}\left (-i \sqrt {15},2 \sqrt {3}\right ) \operatorname {BesselJ}\left (i \sqrt {15},2 \sqrt {3} \sqrt {x}\right )\right )}{\sqrt {3} \left (\operatorname {BesselJ}\left (i \sqrt {15},2 \sqrt {3}\right ) \left (\operatorname {BesselJ}\left (-1-i \sqrt {15},2 \sqrt {3}\right )-\operatorname {BesselJ}\left (1-i \sqrt {15},2 \sqrt {3}\right )\right )+\operatorname {BesselJ}\left (-i \sqrt {15},2 \sqrt {3}\right ) \left (\operatorname {BesselJ}\left (1+i \sqrt {15},2 \sqrt {3}\right )-\operatorname {BesselJ}\left (-1+i \sqrt {15},2 \sqrt {3}\right )\right )\right )} \]