63.1.83 problem 128

Internal problem ID [15523]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 128
Date solved : Thursday, October 02, 2025 at 10:19:32 AM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 59

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

\begin{align*} y &= c_1 \\ y &= \frac {-c_1 c_2 +\sqrt {-2 c_1 \left (-\frac {c_1 \,c_2^{2}}{2}+x +c_3 \right )}}{c_1} \\ y &= \frac {-c_1 c_2 -\sqrt {-2 c_1 \left (-\frac {c_1 \,c_2^{2}}{2}+x +c_3 \right )}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to c_2 \sqrt {2 x+c_1}+c_3 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - 3*Derivative(y(x), (x, 2))**2/Derivative(y(x), (x, 3)) cannot be solved by the factorable group method