Internal problem ID [15490]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 18.2. Expanding a solution in generalized power
series. Bessels equation. Exercises page 177
Problem number: 748.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +4 \left (x^{4}-1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 45
dsolve(x^2*diff(y(x),x$2)-2*x*diff(y(x),x)+4*(x^4-1)*y(x)=0,y(x), singsol=all)
\[ y = -\frac {-\frac {\operatorname {BesselY}\left (\frac {1}{4}, x^{2}\right ) c_{2}}{2}-\frac {\operatorname {BesselJ}\left (\frac {1}{4}, x^{2}\right ) c_{1}}{2}+x^{2} \left (c_{1} \operatorname {BesselJ}\left (-\frac {3}{4}, x^{2}\right )+\operatorname {BesselY}\left (-\frac {3}{4}, x^{2}\right ) c_{2} \right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.158 (sec). Leaf size: 46
DSolve[x^2*y''[x]-2*x*y'[x]+4*(x^4-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^{3/2} \left (c_2 \operatorname {Gamma}\left (\frac {9}{4}\right ) \operatorname {BesselJ}\left (\frac {5}{4},x^2\right )-4 c_1 \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselJ}\left (-\frac {5}{4},x^2\right )\right )}{2^{3/4}} \]