60.3.100 problem 1104

Internal problem ID [11110]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1104
Date solved : Monday, January 27, 2025 at 10:46:00 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 83

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

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

✓ Solution by Mathematica

Time used: 0.070 (sec). Leaf size: 77

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

\[ y(x)\to a^{\frac {1}{2}-\frac {v}{2}} x^{\frac {1}{2}-\frac {v}{2}} \left (c_2 \operatorname {Gamma}(2-v) \operatorname {BesselJ}\left (1-v,2 \sqrt {a} \sqrt {x}\right )+c_1 \operatorname {Gamma}(v) \operatorname {BesselJ}\left (v-1,2 \sqrt {a} \sqrt {x}\right )\right ) \]