Internal problem ID [8841]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1261.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x +2\right ) y^{\prime \prime }+2 \left (n +1+\left (n +1-2 l \right ) x -l \,x^{2}\right ) y^{\prime }+\left (2 l \left (p -n -1\right ) x +2 p l +m \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.484 (sec). Leaf size: 124
dsolve(x*(x+2)*diff(diff(y(x),x),x)+2*(n+1+(n+1-2*l)*x-l*x^2)*diff(y(x),x)+(2*l*(p-n-1)*x+2*p*l+m)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \HeunC \left (4 l , n , n , -4 p l , \frac {\left (4 n +4 p +4\right ) l}{2}-\frac {n^{2}}{2}+m -n , -\frac {x}{2}\right ) \left (x +2\right )^{-\frac {n}{2}-\frac {1}{2}} \left (-\frac {x}{2}-1\right )^{\frac {n}{2}+\frac {1}{2}}+c_{2} \HeunC \left (4 l , -n , n , -4 p l , \frac {\left (4 n +4 p +4\right ) l}{2}-\frac {n^{2}}{2}+m -n , -\frac {x}{2}\right ) \left (x +2\right )^{-\frac {n}{2}-\frac {1}{2}} x^{-n} \left (-\frac {x}{2}-1\right )^{\frac {n}{2}+\frac {1}{2}} \]
✓ Solution by Mathematica
Time used: 0.288 (sec). Leaf size: 120
DSolve[(m + 2*l*p + 2*l*(-1 - n + p)*x)*y[x] + 2*(1 + n + (1 - 2*l + n)*x - l*x^2)*y'[x] + x*(2 + x)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (-\frac {x}{2}-1\right )^{\frac {n+1}{2}} x^{-n} (x+2)^{-\frac {n}{2}-\frac {1}{2}} \left (c_2 \text {HeunC}\left [-4 l n-2 l p-m+n^2+n,-4 l (p-1),1-n,n+1,4 l,-\frac {x}{2}\right ]+c_1 x^n \text {HeunC}\left [-2 l p-m,4 l (n-p+1),n+1,n+1,4 l,-\frac {x}{2}\right ]\right ) \\ \end{align*}