Internal problem ID [10901]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 67.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {x y^{\prime \prime }+a y^{\prime }+y x^{n} b=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 77
dsolve(x*diff(y(x),x$2)+a*diff(y(x),x)+b*x^n*y(x)=0,y(x), singsol=all)
\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (\frac {a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (\frac {a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) \]
✓ Solution by Mathematica
Time used: 0.245 (sec). Leaf size: 165
DSolve[x*y''[x]+a*y'[x]+b*x^n*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (\frac {1}{n}+1\right )^{\frac {a-1}{n+1}} n^{\frac {a-1}{n+1}} b^{\frac {1-a}{2 n+2}} \left (x^n\right )^{-\frac {a-1}{2 n}} \left (c_2 \operatorname {Gamma}\left (\frac {-a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {1-a}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+c_1 \operatorname {Gamma}\left (\frac {a+n}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a-1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )\right ) \]