60.3.256 problem 1261

Internal problem ID [11266]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1261
Date solved : Tuesday, January 28, 2025 at 05:52:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 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{align*}

✓ Solution by Maple

Time used: 0.773 (sec). Leaf size: 106

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)
 

\[ y = \frac {\sqrt {2}\, \sqrt {-x -2}\, \left (x +2\right )^{-\frac {n}{2}-\frac {1}{2}} \left (-\frac {x}{2}-1\right )^{\frac {n}{2}} \left (c_{2} x^{-n} \operatorname {HeunC}\left (4 l , -n , n , -4 p l , 2 \left (1+n +p \right ) l -\frac {n^{2}}{2}+m -n , -\frac {x}{2}\right )+c_{1} \operatorname {HeunC}\left (4 l , n , n , -4 p l , 2 \left (1+n +p \right ) l -\frac {n^{2}}{2}+m -n , -\frac {x}{2}\right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.600 (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)*D[y[x],x] + x*(2 + x)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (-\frac {x}{2}-1\right )^{\frac {n+1}{2}} x^{-n} (x+2)^{-\frac {n}{2}-\frac {1}{2}} \left (c_2 \text {HeunC}\left [-4 l n-2 l p-m+n^2+n,-4 l (p-1),1-n,n+1,4 l,-\frac {x}{2}\right ]+c_1 x^n \text {HeunC}\left [-2 l p-m,4 l (n-p+1),n+1,n+1,4 l,-\frac {x}{2}\right ]\right ) \]