60.3.243 problem 1248

Internal problem ID [11253]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1248
Date solved : Tuesday, January 28, 2025 at 05:52:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.860 (sec). Leaf size: 126

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

\[ y = \frac {{\mathrm e}^{x \sqrt {-b}} \left (2 \operatorname {HeunC}\left (4 \sqrt {-b}, \frac {a}{2}-1, \frac {a}{2}-1, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {1}{2}+\frac {x}{2}\right ) c_{1} +c_{2} \left (-\frac {1}{2}+\frac {x}{2}\right )^{\frac {a}{4}} \left (x +1\right ) \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {a}{4}} \left (x^{2}-1\right )^{-\frac {a}{4}} \operatorname {HeunC}\left (4 \sqrt {-b}, 1-\frac {a}{2}, \frac {a}{2}-1, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {1}{2}+\frac {x}{2}\right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.542 (sec). Leaf size: 192

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

\[ y(x)\to \frac {1}{2} e^{\sqrt {-b} x} \left (c_2 (x-1)^{a/4} \left (x^2-1\right )^{-a/4} (x+1)^{1-\frac {a}{4}} \text {HeunC}\left [\frac {1}{4} a \left (a-4 \sqrt {-b}-2\right )-b+4 \sqrt {-b}+c-d,2 \left (2 \sqrt {-b}+c\right ),2-\frac {a}{2},\frac {a}{2},4 \sqrt {-b},\frac {x+1}{2}\right ]+2 c_1 \text {HeunC}\left [a \sqrt {-b}-b+c-d,2 \left (a \sqrt {-b}+c\right ),\frac {a}{2},\frac {a}{2},4 \sqrt {-b},\frac {x+1}{2}\right ]\right ) \]