75.24.8 problem 748

Internal problem ID [17220]
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 ODEs). Section 18.2. Expanding a solution in generalized power series. Bessels equation. Exercises page 177
Problem number : 748
Date solved : Tuesday, January 28, 2025 at 09:57:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+4 \left (x^{4}-1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (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)
 

\[ y = -\frac {-\frac {\operatorname {BesselY}\left (\frac {1}{4}, x^{2}\right ) c_{2}}{2}-\frac {\operatorname {BesselJ}\left (\frac {1}{4}, x^{2}\right ) c_{1}}{2}+x^{2} \left (\operatorname {BesselJ}\left (-\frac {3}{4}, x^{2}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {3}{4}, x^{2}\right ) c_{2} \right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.144 (sec). Leaf size: 46

DSolve[x^2*D[y[x],{x,2}]-2*x*D[y[x],x]+4*(x^4-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {x^{3/2} \left (c_2 \operatorname {Gamma}\left (\frac {9}{4}\right ) \operatorname {BesselJ}\left (\frac {5}{4},x^2\right )-4 c_1 \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselJ}\left (-\frac {5}{4},x^2\right )\right )}{2^{3/4}} \]