Internal problem ID [9470]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1140.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {a x y^{\prime \prime }+b y^{\prime }+y c=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 134
dsolve(a*x*diff(diff(y(x),x),x)+b*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-\frac {b}{2 a}} \left (-\sqrt {x}\, \operatorname {BesselJ}\left (\frac {a +b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) \sqrt {\frac {c}{a}}\, c_{1} a -\sqrt {x}\, \operatorname {BesselY}\left (\frac {a +b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) \sqrt {\frac {c}{a}}\, c_{2} a +\operatorname {BesselJ}\left (\frac {b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) c_{1} b +\operatorname {BesselY}\left (\frac {b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) c_{2} b \right )}{a \sqrt {\frac {c}{a}}} \]
✓ Solution by Mathematica
Time used: 0.105 (sec). Leaf size: 120
DSolve[c*y[x] + b*y'[x] + a*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to a^{\frac {1}{2} \left (\frac {b}{a}-1\right )} c^{\frac {a-b}{2 a}} x^{\frac {a-b}{2 a}} \left (c_1 \operatorname {Gamma}\left (\frac {b}{a}\right ) \operatorname {BesselJ}\left (\frac {b}{a}-1,\frac {2 \sqrt {c} \sqrt {x}}{\sqrt {a}}\right )+c_2 \operatorname {Gamma}\left (2-\frac {b}{a}\right ) \operatorname {BesselJ}\left (1-\frac {b}{a},\frac {2 \sqrt {c} \sqrt {x}}{\sqrt {a}}\right )\right ) \]