23.4.137 problem 137

Internal problem ID [6439]
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 : 137
Date solved : Tuesday, September 30, 2025 at 02:56:52 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 34

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

\begin{align*} y &= 0 \\ y &= -c_1 \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {-c_1 -c_2 -x}{c_1}}}{c_1}\right )+1\right ) \\ \end{align*}

✓ Mathematica. Time used: 60.061 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to -c_1 \left (1+W\left (-\frac {e^{-\frac {x+c_1+c_2}{c_1}}}{c_1}\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)**2 - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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