2.2.54 Problem 50

Internal problem ID [8857]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 50
Date solved : Friday, April 25, 2025 at 05:15:47 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (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);
 

$$y=-\frac{x}{2}+c_1\mathrm{hypergeom}(\left[\frac{1}{5}\right],[\frac{3}{5},\frac{4}{5}],\frac{x^5}{25})+c_{2}x\mathrm{hypergeom}(\left[\frac{2}{5}\right],[\frac{4}{5},\frac{6}{5}],\frac{x^5}{25})+c_3{x}^2\mathrm{hypergeom}(\left[\frac{3}{5}\right],[\frac{6}{5},\frac{7}{5}],\frac{x^5}{25})$$

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

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & \frac{d}{dx}\frac{d^2}{d{x}^2}y\left(x\right)-x^3\left(\frac{d}{dx}y\left(x\right)\right)-x^{2}y\left(x\right)-x^3=0\\ \bullet & & \text{Highest derivative means the order of the ODE is}3\\ & & \frac{d}{dx}\frac{d^2}{d{x}^2}y\left(x\right)\end{array}$$

✓ 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