Internal problem ID [8981]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1402.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}+\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (a +1\right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 69
dsolve(diff(diff(y(x),x),x) = -1/x/(x^2-1)*((1-4*a)*x^2-1)*diff(y(x),x)-((-v^2+x^2)*(x^2-1)^2+4*a*(a+1)*x^4-2*a*x^2*(x^2-1))/x^2/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{v} \left (x^{2}-1\right )^{a} \left (x^{2}-1\right ) \HeunC \left (0, v , 1, \frac {1}{4}, \frac {a}{2}+\frac {1}{4}, x^{2}\right )+c_{2} x^{-v} \left (x^{2}-1\right )^{a} \left (x^{2}-1\right ) \HeunC \left (0, -v , 1, \frac {1}{4}, \frac {a}{2}+\frac {1}{4}, x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.58 (sec). Leaf size: 80
DSolve[y''[x] == -(((4*a*(1 + a)*x^4 - 2*a*x^2*(-1 + x^2) + (-1 + x^2)^2*(-v^2 + x^2))*y[x])/(x^2*(-1 + x^2)^2)) - ((-1 + (1 - 4*a)*x^2)*y'[x])/(x*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x^2-1\right )^{a+1} x^{-v} \left (c_2 \text {HeunC}\left [-\frac {a}{2}+v-\frac {3}{4},\frac {1}{4},1-v,2,0,x^2\right ]+c_1 x^{2 v} \text {HeunC}\left [\frac {1}{4} (-2 a-4 v-3),\frac {1}{4},v+1,2,0,x^2\right ]\right ) \\ \end{align*}