60.3.14 problem 1014

Internal problem ID [11024]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1014
Date solved : Monday, January 27, 2025 at 10:42:04 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-c \,x^{a} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.705 (sec). Leaf size: 63

dsolve(diff(diff(y(x),x),x)-c*x^a*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.074 (sec). Leaf size: 119

DSolve[-(c*x^a*y[x]) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to (a+2)^{-\frac {1}{a+2}} \sqrt {x} c^{\frac {1}{2 a+4}} \left (c_1 \operatorname {Gamma}\left (\frac {a+1}{a+2}\right ) \operatorname {BesselI}\left (-\frac {1}{a+2},\frac {2 \sqrt {c} x^{\frac {a}{2}+1}}{a+2}\right )+(-1)^{\frac {1}{a+2}} c_2 \operatorname {Gamma}\left (1+\frac {1}{a+2}\right ) \operatorname {BesselI}\left (\frac {1}{a+2},\frac {2 \sqrt {c} x^{\frac {a}{2}+1}}{a+2}\right )\right ) \]