60.3.254 problem 1259

Internal problem ID [11264]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1259
Date solved : Tuesday, January 28, 2025 at 05:52:57 PM
CAS classification : [_Jacobi]

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

✓ Solution by Maple

Time used: 0.258 (sec). Leaf size: 92

dsolve(x*(x-1)*diff(diff(y(x),x),x)+((a+1)*x+b)*diff(y(x),x)-l*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.170 (sec). Leaf size: 111

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

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