Internal problem ID [9855]
Internal file name [OUTPUT/8800_Monday_June_06_2022_05_30_44_AM_51401253/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1531.
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.
\[ \boxed {f^{\prime }\left (x \right ) y^{\prime \prime }+f \left (x \right ) y^{\prime \prime \prime }+g^{\prime }\left (x \right ) y^{\prime }+g \left (x \right ) y^{\prime \prime }+h^{\prime }\left (x \right ) y+h \left (x \right ) y^{\prime }+A \left (x \right ) \left (y^{\prime \prime } f \left (x \right )+g \left (x \right ) y^{\prime }+h \left (x \right ) y\right )=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & f^{\prime }\left (x \right ) y^{\prime \prime }+f \left (x \right ) y^{\prime \prime \prime }+g^{\prime }\left (x \right ) y^{\prime }+g \left (x \right ) y^{\prime \prime }+h^{\prime }\left (x \right ) y+h \left (x \right ) y^{\prime }+A \left (x \right ) \left (y^{\prime \prime } f \left (x \right )+g \left (x \right ) y^{\prime }+h \left (x \right ) y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \end {array} \]
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 -> Try computing a Rational Normal Form for the given ODE... <- unable to resolve the Equivalence to a Rational Normal Form trying reduction of order using simple exponentials trying differential order: 3; exact nonlinear --- Trying Lie symmetry methods, high order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
dsolve(diff(f(x),x)*diff(diff(y(x),x),x)+f(x)*diff(diff(diff(y(x),x),x),x)+diff(g(x),x)*diff(y(x),x)+g(x)*diff(diff(y(x),x),x)+diff(h(x),x)*y(x)+h(x)*diff(y(x),x)+A(x)*(f(x)*diff(diff(y(x),x),x)+g(x)*diff(y(x),x)+h(x)*y(x))=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*Derivative[1][h][x] + h[x]*y'[x] + Derivative[1][g][x]*y'[x] + g[x]*y''[x] + Derivative[1][f][x]*y''[x] + A[x]*(h[x]*y[x] + g[x]*y'[x] + f[x]*y''[x]) + f[x]*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved