Internal problem ID [9787]
Internal file name [OUTPUT/8730_Monday_June_06_2022_05_21_08_AM_14376527/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1461.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime }-\left (3 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime }+\left (b +c \operatorname {JacobiSN}\left (z , x\right )^{2}-3 k^{2} \operatorname {JacobiSN}\left (z , x\right ) \operatorname {JacobiCN}\left (z , x\right ) \operatorname {JacobiDN}\left (z , x\right )\right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-\left (3 k^{2} \mathrm {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime }+\left (b +c \mathrm {JacobiSN}\left (z , x\right )^{2}-3 k^{2} \mathrm {JacobiSN}\left (z , x\right ) \mathrm {JacobiCN}\left (z , x\right ) \mathrm {JacobiDN}\left (z , x\right )\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 3; missing the dependent variable -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius trying a solution in terms of MeijerG functions trying reduction of order using simple exponentials -> trying with_periodic_functions in the coefficients --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 5 --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(diff(diff(diff(y(x),x),x),x)-(3*k^2*JacobiSN(z,x)^2+a)*diff(y(x),x)+(b+c*JacobiSN(z,x)^2-3*k^2*JacobiSN(z,x)*JacobiCN(z,x)*JacobiDN(z,x))*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(b - 3*k^2*JacobiCN[z, x]*JacobiDN[z, x]*JacobiSN[z, x] + c*JacobiSN[z, x]^2)*y[x] - (a + 3*k^2*JacobiSN[z, x]^2)*y'[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved