Internal problem ID [10716]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {x^{2} z^{\prime \prime }+3 z^{\prime } x +4 z=0} \] 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)
\[ z \left (x \right ) = \frac {5 \sqrt {3}\, \sin \left (\sqrt {3}\, \ln \left (x \right )\right )}{3 x} \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 171
DSolve[{x^2*z''[x]+3*x*z[x]+4*z[x]==0,{z[1]==0,z'[1]==5}},z[x],x,IncludeSingularSolutions -> True]
\begin{align*} z(x)\to \frac {5 \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 )}{\, _0\tilde {F}_1\left (;i \sqrt {15};-3\right ) \, _0\tilde {F}_1\left (;1-i \sqrt {15};-3\right )-\left (\, _0\tilde {F}_1\left (;-i \sqrt {15};-3\right )+i \sqrt {15} \, _0\tilde {F}_1\left (;1-i \sqrt {15};-3\right )\right ) \, _0\tilde {F}_1\left (;1+i \sqrt {15};-3\right )} \\ \end{align*}