Internal problem ID [9584]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1249.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+y c=0} \]
✓ Solution by Maple
Time used: 0.094 (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)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}, -\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}\right ], \left [\frac {a}{2}-\frac {b}{2}\right ], \frac {x}{2}+\frac {1}{2}\right )+c_{2} \left (\frac {x}{2}+\frac {1}{2}\right )^{1-\frac {a}{2}+\frac {b}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {b}{2}, \frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {b}{2}\right ], \left [2-\frac {a}{2}+\frac {b}{2}\right ], \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.204 (sec). Leaf size: 190
DSolve[c*y[x] + (b + a*x)*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} (x-1)^{\frac {1}{2} (-a-b)} \left (2 c_1 (x-1)^{\frac {a+b}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (a-\sqrt {a^2-2 a-4 c+1}-1\right ),\frac {1}{2} \left (a+\sqrt {a^2-2 a-4 c+1}-1\right ),\frac {a+b}{2},\frac {1-x}{2}\right )+c_2 (x-1) 2^{\frac {a+b}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-b-\sqrt {a^2-2 a-4 c+1}+1\right ),\frac {1}{2} \left (-b+\sqrt {a^2-2 a-4 c+1}+1\right ),\frac {1}{2} (-a-b+4),\frac {1-x}{2}\right )\right ) \]