61.29.14 problem 123

Internal problem ID [12623]
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
Problem number : 123
Date solved : Tuesday, January 28, 2025 at 03:23:23 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }+a x y^{\prime }+b y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 57

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

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