23.6.7 problem 7

Internal problem ID [6806]
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 : 7
Date solved : Friday, October 03, 2025 at 02:09:53 AM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

\begin{align*} 3 y^{\prime } y^{\prime \prime }+\left (a +y\right ) y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.081 (sec). Leaf size: 95

ode:=3*diff(y(x),x)*diff(diff(y(x),x),x)+(a+y(x))*diff(diff(diff(y(x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -a \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (-2 \int _{}^{\textit {\_Z}}\frac {1}{-4 \textit {\_h}^{2}+\sqrt {c_1 \left (-4 \textit {\_h}^{2}+c_1 \right )}+c_1}d \textit {\_h} +x +c_2 \right )d x +c_3}-a \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (2 \int _{}^{\textit {\_Z}}-\frac {1}{-4 \textit {\_h}^{2}-\sqrt {c_1 \left (-4 \textit {\_h}^{2}+c_1 \right )}+c_1}d \textit {\_h} +x +c_2 \right )d x +c_3}-a \\ \end{align*}

✓ Mathematica. Time used: 0.101 (sec). Leaf size: 83

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

\begin{align*} y(x)&\to -\frac {a c_1+\sqrt {c_1{}^3 (x+c_3){}^2-e^{2 c_2} c_1}}{c_1}\\ y(x)&\to \frac {\sqrt {c_1{}^3 (x+c_3){}^2-e^{2 c_2} c_1}-a c_1}{c_1} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(-a - y(x))*Derivative(y(x), (x, 3))/(3*Derivative(y(x), (x, 2)