23.6.1 problem 1

Internal problem ID [6800]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 6. THE EQUATION IS NONLINEAR AND OF ORDER GREATER THAN TWO, page 427
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 03:51:47 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]

\begin{align*} y^{\prime \prime \prime }&=y^{\prime } \left (1+y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 67

ode:=diff(diff(diff(y(x),x),x),x) = diff(y(x),x)*(1+diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \int \operatorname {RootOf}\left (3 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {6 \textit {\_f}^{3}+9 \textit {\_f}^{2}+9 c_1}}d \textit {\_f} +x +c_2 \right )d x +c_3 \\ y &= \int \operatorname {RootOf}\left (-3 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {6 \textit {\_f}^{3}+9 \textit {\_f}^{2}+9 c_1}}d \textit {\_f} +x +c_2 \right )d x +c_3 \\ \end{align*}

✗ Mathematica

ode=D[y[x],{x,3}] == D[y[x],x]*(1 + D[y[x],x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(Derivative(y(x), x) + 1)*Derivative(y(x), x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(4*Derivative(y(x), (x, 3)) + 1)/2 + Derivative(y(x), x) +