60.3.93 problem 1097

Internal problem ID [11103]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1097
Date solved : Monday, January 27, 2025 at 10:45:48 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }-y^{\prime }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 70

dsolve(x*diff(diff(y(x),x),x)-diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right ) \sqrt {x}\, c_{1} +\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right ) \sqrt {x}\, c_{2} -x \sqrt {a}\, \left (c_{1} \operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )}{\sqrt {a}} \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 45

DSolve[a*y[x] - D[y[x],x] + x*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 2 a x \left (c_1 \operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )-c_2 \operatorname {BesselY}\left (2,2 \sqrt {a} \sqrt {x}\right )\right ) \]