60.3.244 problem 1249

Internal problem ID [11254]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1249
Date solved : Tuesday, January 28, 2025 at 05:52:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.181 (sec). Leaf size: 134

dsolve((x^2-1)*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.193 (sec). Leaf size: 190

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

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