60.3.99 problem 1103

Internal problem ID [11109]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1103
Date solved : Monday, January 27, 2025 at 10:45:59 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }-2 y^{\prime }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 80

dsolve(x*diff(diff(y(x),x),x)-2*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 56

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

\[ y(x)\to 2 a^{3/2} x^{3/2} \left (3 c_1 \operatorname {BesselJ}\left (3,2 \sqrt {a} \sqrt {x}\right )-i c_2 \operatorname {BesselY}\left (3,2 \sqrt {a} \sqrt {x}\right )\right ) \]