Internal problem ID [10886]
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: 62.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {x y^{\prime \prime }+a y^{\prime }+y b=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 83
dsolve(x*diff(y(x),x$2)+a*diff(y(x),x)+b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-\sqrt {x}\, \operatorname {BesselJ}\left (a +1, 2 \sqrt {b}\, \sqrt {x}\right ) \sqrt {b}\, c_{1} -\sqrt {x}\, \operatorname {BesselY}\left (a +1, 2 \sqrt {b}\, \sqrt {x}\right ) \sqrt {b}\, c_{2} +a \left (\operatorname {BesselJ}\left (a , 2 \sqrt {b}\, \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (a , 2 \sqrt {b}\, \sqrt {x}\right ) c_{2} \right )\right ) x^{-\frac {a}{2}}}{\sqrt {b}} \]
✓ Solution by Mathematica
Time used: 0.111 (sec). Leaf size: 77
DSolve[x*y''[x]+a*y'[x]+b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to b^{\frac {1}{2}-\frac {a}{2}} x^{\frac {1}{2}-\frac {a}{2}} \left (c_2 \operatorname {Gamma}(2-a) \operatorname {BesselJ}\left (1-a,2 \sqrt {b} \sqrt {x}\right )+c_1 \operatorname {Gamma}(a) \operatorname {BesselJ}\left (a-1,2 \sqrt {b} \sqrt {x}\right )\right ) \]