60.3.328 problem 1334

Internal problem ID [11338]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1334
Date solved : Tuesday, January 28, 2025 at 06:02:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (\left (a +1\right ) x -1\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (x -1\right )} \end{align*}

✓ Solution by Maple

Time used: 0.260 (sec). Leaf size: 89

dsolve(diff(diff(y(x),x),x) = -((a+1)*x-1)/x/(x-1)*diff(y(x),x)-1/4*((a^2-b^2)*x+c^2)/x^2/(x-1)*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.227 (sec). Leaf size: 89

DSolve[D[y[x],{x,2}] == -1/4*((c^2 + (a^2 - b^2)*x)*y[x])/((-1 + x)*x^2) - ((-1 + (1 + a)*x)*D[y[x],x])/((-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
 

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