Internal problem ID [8594]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1014.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-c \,x^{a} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 65
dsolve(diff(diff(y(x),x),x)-c*x^a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}\, \BesselJ \left (\frac {1}{2+a}, \frac {2 \sqrt {-c}\, x^{1+\frac {a}{2}}}{2+a}\right )+c_{2} \sqrt {x}\, \BesselY \left (\frac {1}{2+a}, \frac {2 \sqrt {-c}\, x^{1+\frac {a}{2}}}{2+a}\right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 119
DSolve[-(c*x^a*y[x]) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (a+2)^{-\frac {1}{a+2}} \sqrt {x} c^{\frac {1}{2 a+4}} \left (c_1 \text {Gamma}\left (\frac {a+1}{a+2}\right ) I_{-\frac {1}{a+2}}\left (\frac {2 \sqrt {c} x^{\frac {a}{2}+1}}{a+2}\right )+(-1)^{\frac {1}{a+2}} c_2 \text {Gamma}\left (\frac {1}{a+2}+1\right ) I_{\frac {1}{a+2}}\left (\frac {2 \sqrt {c} x^{\frac {a}{2}+1}}{a+2}\right )\right ) \\ \end{align*}