65.12.5 problem 19.1 (v)

Internal problem ID [13721]
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 : Wednesday, March 05, 2025 at 10:13:51 PM
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*}

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

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

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

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 220

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

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

✓ Sympy. Time used: 0.204 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
z = Function("z") 
ode = Eq(x**2*Derivative(z(x), (x, 2)) + 3*x*Derivative(z(x), x) + 4*z(x),0) 
ics = {z(1): 0, Subs(Derivative(z(x), x), x, 1): 5} 
dsolve(ode,func=z(x),ics=ics)
 

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