Internal problem ID [9777]
Internal file name [OUTPUT/8720_Monday_June_06_2022_05_19_56_AM_6607751/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1451.
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.
\[ \boxed {y^{\prime \prime \prime }-a \,x^{b} y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }-a \,x^{b} y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
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 <- MeijerG function solution successful`
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 114
dsolve(diff(diff(diff(y(x),x),x),x)-a*x^b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +1}{b +3}, \frac {b +2}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +2}{b +3}, \frac {4+b}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +5}{b +3}, \frac {4+b}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right ) \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 164
DSolve[-(a*x^b*y[x]) + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{\frac {1}{b+3}} (b+3)^{-\frac {6}{b+3}} x a^{\frac {1}{b+3}} \left ((-1)^{\frac {1}{b+3}} c_3 x a^{\frac {1}{b+3}} \, _0F_2\left (;1+\frac {1}{b+3},1+\frac {2}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right )+(b+3)^{\frac {3}{b+3}} c_2 \, _0F_2\left (;1-\frac {1}{b+3},1+\frac {1}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right )\right )+c_1 \, _0F_2\left (;1-\frac {2}{b+3},1-\frac {1}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right ) \]