60.3.312 problem 1318

Internal problem ID [11322]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1318
Date solved : Tuesday, January 28, 2025 at 05:59:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.300 (sec). Leaf size: 122

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

\[ y = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{4}+\frac {a}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, -\frac {1}{4}+\frac {a}{4}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}\right ], \left [-\frac {b}{2}+\frac {1}{2}\right ], x^{2}\right )+c_{2} x^{b +1} \operatorname {hypergeom}\left (\left [\frac {1}{4}+\frac {a}{4}+\frac {b}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, \frac {1}{4}+\frac {a}{4}+\frac {b}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}\right ], \left [\frac {3}{2}+\frac {b}{2}\right ], x^{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.320 (sec). Leaf size: 146

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

\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (a-\sqrt {a^2-2 a-4 c+1}-1\right ),\frac {1}{4} \left (a+\sqrt {a^2-2 a-4 c+1}-1\right ),\frac {1-b}{2},x^2\right )+i^{b+1} c_2 x^{b+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (a+2 b-\sqrt {a^2-2 a-4 c+1}+1\right ),\frac {1}{4} \left (a+2 b+\sqrt {a^2-2 a-4 c+1}+1\right ),\frac {b+3}{2},x^2\right ) \]