60.3.319 problem 1325

Internal problem ID [11329]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1325
Date solved : Tuesday, January 28, 2025 at 05:59:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.240 (sec). Leaf size: 86

dsolve(diff(diff(y(x),x),x) = -((a+b+1)*x+alpha+beta-1)/x/(x-1)*diff(y(x),x)-(a*b*x-alpha*beta)/x^2/(x-1)*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.262 (sec). Leaf size: 52

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

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