2.2.54 Problem 50

Internal problem ID [8858]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 50
Date solved : Thursday, March 13, 2025 at 06:27:40 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3}&=0 \end{align*}

Does not support this form of ODE for higher order. Terminating.

✓ Maple. Time used: 0.010 (sec). Leaf size: 51

ode:=diff(diff(diff(y(x),x),x),x)-x^3*diff(y(x),x)-x^2*y(x)-x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x}{2}+c_{1} \operatorname {hypergeom}\left (\left [\frac {1}{5}\right ], \left [\frac {3}{5}, \frac {4}{5}\right ], \frac {x^{5}}{25}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\frac {2}{5}\right ], \left [\frac {4}{5}, \frac {6}{5}\right ], \frac {x^{5}}{25}\right )+c_3 \,x^{2} \operatorname {hypergeom}\left (\left [\frac {3}{5}\right ], \left [\frac {6}{5}, \frac {7}{5}\right ], \frac {x^{5}}{25}\right ) \]

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 `
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right )-\left (\frac {d}{d x}y \left (x \right )\right ) x^{3}-x^{2} y \left (x \right )-x^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right ) \end {array} \]

✓ Mathematica. Time used: 10.995 (sec). Leaf size: 2548

ode=D[y[x],{x,3}]-x^3*D[y[x],x]-x^2*y[x]-x^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*Derivative(y(x), x) - x**3 - x**2*y(x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) + 1 + y(x)/x - Derivative(y(x), (x, 3))/x**3 cannot be solved by the factorable group method