66.2.22 problem Problem 31

Internal problem ID [13850]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 31
Date solved : Monday, March 31, 2025 at 08:15:10 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.339 (sec). Leaf size: 42

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

\begin{align*} y &= -\frac {4}{-4 c_1 +x} \\ y &= c_1 \\ y &= {\mathrm e}^{-c_1 \left (c_2 +x \right )}-c_1 \\ y &= {\mathrm e}^{c_1 \left (c_2 +x \right )}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 8.33 (sec). Leaf size: 119

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

\begin{align*} y(x)\to \frac {1}{2} \left (e^{-\frac {1}{2} \left (1+e^{c_1}\right ) (x+c_2)}-1-e^{c_1}\right ) \\ y(x)\to \frac {1+e^{\frac {x+c_2}{-1+\tanh \left (\frac {c_1}{2}\right )}}}{-1+\tanh \left (\frac {c_1}{2}\right )} \\ y(x)\to -\frac {1}{2}-\frac {1}{2} e^{-\frac {x}{2}-\frac {c_2}{2}} \\ y(x)\to \frac {1}{2} \left (-1+e^{-\frac {x}{2}-\frac {c_2}{2}}\right ) \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE (sqrt(-108*y(x)**3*Derivative(y(x), (x, 2))**3 + 729*Derivative(y(x), (x, 2))**4)/2 + 27*Derivative(y(x), (x, 2))**2/2)**(1/3)/3 + Derivative(y(x), x) + y(x)*Derivative(y(x), (x, 2))/(sqrt(-108*y(x)**3*Derivative(y(x), (x, 2))**3 + 729*Derivative(y(x), (x, 2))**4)/2 + 27*Derivative(y(x), (x, 2))**2/2)**(1/3) cannot be solved by the factorable group method