60.3.366 problem 1372

Internal problem ID [11376]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1372
Date solved : Tuesday, January 28, 2025 at 06:04:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 6.653 (sec). Leaf size: 101

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

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

✓ Solution by Mathematica

Time used: 0.808 (sec). Leaf size: 189

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

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