Internal problem ID [8607]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1027.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \mathrm {sn}\left (x | k \right )^{2}+b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.166 (sec). Leaf size: 69
dsolve(diff(diff(y(x),x),x)-(n*(n+1)*k^2*JacobiSN(x,k)^2+b)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \mathit {HG}\left (\frac {1}{k^{2}}, \frac {b}{4 k^{2}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \mathrm {sn}\left (x | k \right )^{2}\right )+c_{2} \mathit {HG}\left (\frac {1}{k^{2}}, \frac {k^{2}+b +1}{4 k^{2}}, 1+\frac {n}{2}, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \mathrm {sn}\left (x | k \right )^{2}\right ) \mathrm {sn}\left (x | k \right ) \]
✓ Solution by Mathematica
Time used: 0.666 (sec). Leaf size: 167
DSolve[(b + a*JacobiSN[x, k]^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {-\text {dn}(x|k)^2} \left (c_1 \text {HeunG}\left [\frac {1}{k},\frac {k-b}{4 k},\frac {1}{4} \left (\frac {\sqrt {k-4 a}}{\sqrt {k}}+3\right ),\frac {3}{4}-\frac {\sqrt {k-4 a}}{4 \sqrt {k}},\frac {1}{2},\frac {1}{2},\text {sn}(x|k)^2\right ]+c_2 \text {sn}(x|k) \text {HeunG}\left [\frac {1}{k},1-\frac {b-1}{4 k},\frac {1}{4} \left (\frac {\sqrt {k-4 a}}{\sqrt {k}}+5\right ),\frac {5}{4}-\frac {\sqrt {k-4 a}}{4 \sqrt {k}},\frac {3}{2},\frac {1}{2},\text {sn}(x|k)^2\right ]\right ) \\ \end{align*}