61.30.23 problem 171

Internal problem ID [12671]
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 : 171
Date solved : Tuesday, January 28, 2025 at 08:10:39 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (x -1\right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.365 (sec). Leaf size: 44

dsolve(x*(x-1)*diff(y(x),x$2)+((alpha+beta+1)*x-gamma)*diff(y(x),x)+alpha*beta*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {hypergeom}\left (\left [\alpha , \beta \right ], \left [\gamma \right ], x\right )+c_{2} x^{1-\gamma } \operatorname {hypergeom}\left (\left [\beta -\gamma +1, \alpha -\gamma +1\right ], \left [-\gamma +2\right ], x\right ) \]

✓ Solution by Mathematica

Time used: 0.184 (sec). Leaf size: 49

DSolve[x*(x-1)*D[y[x],{x,2}]+((\[Alpha]+\[Beta]+1)*x-\[Gamma])*D[y[x],x]+\[Alpha]*\[Beta]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}(\alpha ,\beta ,\gamma ,x)-(-1)^{-\gamma } c_2 x^{1-\gamma } \operatorname {Hypergeometric2F1}(\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,x) \]