Internal problem ID [9439]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1104.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {y^{\prime \prime } x +v y^{\prime }+a y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 47
dsolve(x*diff(diff(y(x),x),x)+v*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{\frac {1}{2}-\frac {v}{2}} \operatorname {BesselJ}\left (-1+v , 2 \sqrt {a}\, \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {v}{2}} \operatorname {BesselY}\left (-1+v , 2 \sqrt {a}\, \sqrt {x}\right ) \]
✓ Solution by Mathematica
Time used: 0.07 (sec). Leaf size: 77
DSolve[a*y[x] + v*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to a^{\frac {1}{2}-\frac {v}{2}} x^{\frac {1}{2}-\frac {v}{2}} \left (c_2 \operatorname {Gamma}(2-v) \operatorname {BesselJ}\left (1-v,2 \sqrt {a} \sqrt {x}\right )+c_1 \operatorname {Gamma}(v) \operatorname {BesselJ}\left (v-1,2 \sqrt {a} \sqrt {x}\right )\right ) \]