61.30.15 problem 163

Internal problem ID [12663]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 163
Date solved : Tuesday, January 28, 2025 at 03:29:55 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}+a \right ) y^{\prime \prime }+2 b x y^{\prime }+2 \left (b -1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.190 (sec). Leaf size: 41

dsolve((x^2+a)*diff(y(x),x$2)+2*b*x*diff(y(x),x)+2*(b-1)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \left (\frac {x^{2}+a}{a}\right )^{1-b}+c_{2} x \operatorname {hypergeom}\left (\left [1, b -\frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {x^{2}}{a}\right ) \]

✓ Solution by Mathematica

Time used: 0.274 (sec). Leaf size: 64

DSolve[(x^2+a)*D[y[x],{x,2}]+2*b*x*D[y[x],x]+2*(b-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (a+x^2\right ) \left (\frac {c_2 x \left (\frac {a+x^2}{a}\right )^{-b} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2-b,\frac {3}{2},-\frac {x^2}{a}\right )}{a^2}+c_1 \left (a+x^2\right )^{-b}\right ) \]