Internal problem ID [8720]
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]]
Solve \begin {gather*} \boxed {a x y^{\prime \prime }+y^{\prime } b +c y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 77
dsolve(a*x*diff(diff(y(x),x),x)+b*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{\frac {a -b}{2 a}} \BesselJ \left (\frac {-a +b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right )+c_{2} x^{\frac {a -b}{2 a}} \BesselY \left (\frac {-a +b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 120
DSolve[c*y[x] + b*y'[x] + a*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 \text {Gamma}\left (\frac {b}{a}\right ) J_{\frac {b}{a}-1}\left (\frac {2 \sqrt {c} \sqrt {x}}{\sqrt {a}}\right )+c_2 \text {Gamma}\left (2-\frac {b}{a}\right ) J_{1-\frac {b}{a}}\left (\frac {2 \sqrt {c} \sqrt {x}}{\sqrt {a}}\right )\right ) \\ \end{align*}