Internal problem ID [9362]
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]]
\[ \boxed {y^{\prime \prime }-\left (n \left (1+n \right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.859 (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)
\[ y \left (x \right ) = c_{1} \operatorname {HeunG}\left (\frac {1}{k^{2}}, \frac {b}{4 k^{2}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \operatorname {JacobiSN}\left (x , k\right )^{2}\right )+c_{2} \operatorname {HeunG}\left (\frac {1}{k^{2}}, \frac {k^{2}+b +1}{4 k^{2}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \operatorname {JacobiSN}\left (x , k\right )^{2}\right ) \operatorname {JacobiSN}\left (x , k\right ) \]
✓ Solution by Mathematica
Time used: 1.268 (sec). Leaf size: 209
DSolve[(b + a*JacobiSN[x, k]^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {k \text {sn}(x|k)^2-1} \left (c_1 \text {HeunG}\left [\frac {1}{k},\frac {k-b}{4 k},\frac {1}{4} \left (\frac {\sqrt {k-4 a}}{\sqrt {k}}+3\right ),\frac {\sqrt {k} \sqrt {k-4 a}+2 a+k}{2 \left (\sqrt {k} \sqrt {k-4 a}+k\right )},\frac {1}{2},\frac {1}{2},\text {sn}(x|k)^2\right ]+c_2 \text {sn}(x|k) \text {HeunG}\left [\frac {1}{k},\frac {-b+4 k+1}{4 k},\frac {1}{4} \left (\frac {\sqrt {k-4 a}}{\sqrt {k}}+5\right ),\frac {\sqrt {k} \sqrt {k-4 a}+a+k}{\sqrt {k} \sqrt {k-4 a}+k},\frac {3}{2},\frac {1}{2},\text {sn}(x|k)^2\right ]\right ) \]