Internal problem ID [9360]
Internal file name [OUTPUT/8296_Monday_June_06_2022_02_39_44_AM_61249719/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunG ODE, case a <> 0, e <> 0, g <> 0, c = 0 Change of variables used: [x = InverseJacobiSN(t,k)] Linear ODE actually solved: (-k^2*n^2*t^2-k^2*n*t^2-b)*u(t)+(2*k^2*t^3-k^2*t-t)*diff(u(t),t)+(k^2*t^4-k^2*t^2-t^2+1)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 1.282 (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]
\[ y(x)\to \sqrt {k \text {sn}(x|k)^2-1} \left (c_1 \text {HeunG}\left [\frac {1}{k},\frac {k-b}{4 k},\frac {1}{4} \left (\frac {\sqrt {k-4 a}}{\sqrt {k}}+3\right ),\frac {\sqrt {k} \sqrt {k-4 a}+2 a+k}{2 \left (\sqrt {k} \sqrt {k-4 a}+k\right )},\frac {1}{2},\frac {1}{2},\text {sn}(x|k)^2\right ]+c_2 \text {sn}(x|k) \text {HeunG}\left [\frac {1}{k},\frac {-b+4 k+1}{4 k},\frac {1}{4} \left (\frac {\sqrt {k-4 a}}{\sqrt {k}}+5\right ),\frac {\sqrt {k} \sqrt {k-4 a}+a+k}{\sqrt {k} \sqrt {k-4 a}+k},\frac {3}{2},\frac {1}{2},\text {sn}(x|k)^2\right ]\right ) \]