23.6.6 problem 6

Internal problem ID [6805]
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 : 6
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]]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 81

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

\begin{align*} y &= 0 \\ -2 \int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{4}+4 c_2 \,\textit {\_a}^{2}-4 c_2^{2}+4 c_1}}d \textit {\_a} -x -c_3 &= 0 \\ 2 \int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{4}+4 c_2 \,\textit {\_a}^{2}-4 c_2^{2}+4 c_1}}d \textit {\_a} -x -c_3 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.791 (sec). Leaf size: 409

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {2 i \sqrt {1+\frac {\text {$\#$1}^2}{2 \left (\sqrt {c_2{}^2-c_1}-c_2\right )}} \sqrt {1-\frac {\text {$\#$1}^2}{2 \left (c_2+\sqrt {c_2{}^2-c_1}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \text {$\#$1}}{\sqrt {2}}\right ),\frac {c_2-\sqrt {c_2{}^2-c_1}}{c_2+\sqrt {c_2{}^2-c_1}}\right )}{\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \sqrt {-\frac {\text {$\#$1}^4}{2}+2 \text {$\#$1}^2 c_2-2 c_1}}\&\right ][x+c_3]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 i \sqrt {1+\frac {\text {$\#$1}^2}{2 \left (\sqrt {c_2{}^2-c_1}-c_2\right )}} \sqrt {1-\frac {\text {$\#$1}^2}{2 \left (c_2+\sqrt {c_2{}^2-c_1}\right )}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \text {$\#$1}}{\sqrt {2}}\right ),\frac {c_2-\sqrt {c_2{}^2-c_1}}{c_2+\sqrt {c_2{}^2-c_1}}\right )}{\sqrt {\frac {1}{\sqrt {c_2{}^2-c_1}-c_2}} \sqrt {-\frac {\text {$\#$1}^4}{2}+2 \text {$\#$1}^2 c_2-2 c_1}}\&\right ][x+c_3] \end{align*}

✗ Sympy

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

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