Internal problem ID [11005]
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 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 171.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
\[ \boxed {x \left (x -1\right ) y^{\prime \prime }+\left (\left (\beta +\alpha +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y=0} \]
✓ Solution by Maple
Time used: 0.046 (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)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\alpha , \beta \right ], \left [\gamma \right ], x\right )+c_{2} x^{1-\gamma } \operatorname {hypergeom}\left (\left [\beta +1-\gamma , \alpha +1-\gamma \right ], \left [2-\gamma \right ], x\right ) \]
✓ Solution by Mathematica
Time used: 0.281 (sec). Leaf size: 49
DSolve[x*(x-1)*y''[x]+((\[Alpha]+\[Beta]+1)*x-\[Gamma])*y'[x]+\[Alpha]*\[Beta]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}(\alpha ,\beta ,\gamma ,x)-(-1)^{-\gamma } c_2 x^{1-\gamma } \operatorname {Hypergeometric2F1}(\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,x) \]