29.14 problem 123

Internal problem ID [10957]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form \(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 123.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_Emden, _Fowler]]

\[ \boxed {x^{2} y^{\prime \prime }+a x y^{\prime }+b y=0} \]

Solution by Maple

Time used: 0.0 (sec). Leaf size: 53

dsolve(x^2*diff(y(x),x$2)+a*x*diff(y(x),x)+b*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} x^{-\frac {a}{2}+\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}}+c_{2} x^{-\frac {a}{2}+\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \]

Solution by Mathematica

Time used: 0.069 (sec). Leaf size: 57

DSolve[x^2*y''[x]+a*x*y'[x]+b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 b+1}-a+1\right )} \left (c_2 x^{\sqrt {a^2-2 a-4 b+1}}+c_1\right ) \]