61.30.20 problem 168

Internal problem ID [12668]
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 : 168
Date solved : Tuesday, January 28, 2025 at 08:10:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.306 (sec). Leaf size: 134

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

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

✓ Solution by Mathematica

Time used: 0.194 (sec). Leaf size: 184

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

\[ y(x)\to 2^{\frac {1}{2} (-a-b-2)} \left (c_2 (x-1)^{\frac {1}{2} (a+b+2)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (b-\sqrt {a^2+2 a+4 c+1}+1\right ),\frac {1}{2} \left (b+\sqrt {a^2+2 a+4 c+1}+1\right ),\frac {1}{2} (a+b+4),\frac {1-x}{2}\right )+c_1 2^{\frac {1}{2} (a+b+2)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-a-\sqrt {a^2+2 a+4 c+1}-1\right ),\frac {1}{2} \left (-a+\sqrt {a^2+2 a+4 c+1}-1\right ),\frac {1}{2} (-a-b),\frac {1-x}{2}\right )\right ) \]