23.4.161 problem 161

Internal problem ID [6463]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 161
Date solved : Tuesday, September 30, 2025 at 03:01:30 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 36

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

\begin{align*} y &= 0 \\ y &= c_1 \\ y &= {\mathrm e}^{\frac {-c_1 \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {c_2 +x}{c_1}}}{c_1}\right )+c_2 +x}{c_1}} \\ \end{align*}

✓ Mathematica. Time used: 12.051 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to e^{c_1} W\left (e^{e^{-c_1} \left (x-e^{c_1} c_1+c_2\right )}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE (sqrt((27*y(x)*Derivative(y(x), (x, 2)) - 2)**2 - 4)/2 + 27*y(x)