Internal problem ID [7190]
Internal file name [OUTPUT/6176_Sunday_June_05_2022_04_26_49_PM_70486816/index.tex]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 50.
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.
\[ \boxed {y^{\prime \prime \prime }-x^{3} y^{\prime }-x^{2} y=x^{3}} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }-x^{3} y^{\prime }-x^{2} y=x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 3; missing the dependent variable checking if the LODE is of Euler type trying Louvillian solutions for 3rd order ODEs, imprimitive case -> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under a power @ Moebius <- pFq successful: received ODE is equivalent to the 1F2 ODE, case c = 0 `
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 51
dsolve(diff(y(x),x$3)-x^3*diff(y(x),x)-x^2*y(x)-x^3=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {x}{2}+c_{1} \operatorname {hypergeom}\left (\left [\frac {1}{5}\right ], \left [\frac {3}{5}, \frac {4}{5}\right ], \frac {x^{5}}{25}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\frac {2}{5}\right ], \left [\frac {4}{5}, \frac {6}{5}\right ], \frac {x^{5}}{25}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\frac {3}{5}\right ], \left [\frac {6}{5}, \frac {7}{5}\right ], \frac {x^{5}}{25}\right ) \]
✓ Solution by Mathematica
Time used: 12.206 (sec). Leaf size: 2548
DSolve[y'''[x]-x^3*y'[x]-x^2*y[x]-x^3==0,y[x],x,IncludeSingularSolutions -> True]
Too large to display