Internal problem ID [9866]
Internal file name [OUTPUT/8811_Monday_June_06_2022_05_32_16_AM_71772591/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1542.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime \prime }+a \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime \prime }+b \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\left (c \left (6 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )^{2}-\frac {\operatorname {g2}}{2}\right )+d \right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }+a \mathit {WeierstrassP}\left (x , \mathit {g2} , \mathit {g3}\right ) \left (\frac {d}{d x}y^{\prime }\right )+b \mathit {WeierstrassPPrime}\left (x , \mathit {g2} , \mathit {g3}\right ) y^{\prime }+\left (c \left (6 \mathit {WeierstrassP}\left (x , \mathit {g2} , \mathit {g3}\right )^{2}-\frac {\mathit {g2}}{2}\right )+d \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for high 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: 4; missing the dependent variable trying a solution in terms of MeijerG functions trying a solution in terms of MeijerG functions 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`[0, y]
✗ Solution by Maple
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)+a*WeierstrassP(x,g2,g3)*diff(diff(y(x),x),x)+b*WeierstrassPPrime(x,g2,g3)*diff(y(x),x)+(c*(6*WeierstrassP(x,g2,g3)^2-1/2*g2)+d)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(d + c*(-1/2*g2 + 6*WeierstrassP[x, {g2, g3}]^2))*y[x] + b*WeierstrassPPrime[x, {g2, g3}]*y'[x] + a*WeierstrassP[x, {g2, g3}]*y''[x] + Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved