69.1.78 problem 123

Internal problem ID [14160]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 123
Date solved : Monday, March 31, 2025 at 12:11:11 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.141 (sec). Leaf size: 5

ode:=y(x)*diff(diff(y(x),x),x)-diff(y(x),x)^2+diff(y(x),x)^3 = 0; 
ic:=y(0) = -1, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -1 \]

✗ Mathematica

ode=y[x]*D[y[x],{x,2}]-(D[y[x],x])^2+(D[y[x],x])^3==0; 
ic={y[0]==-1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✗ 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 = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 0} 
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)*Derivative(y(x), (x, 2))/2 - 1)**(1/3)/3 + Derivative(y(x), x) - 1/3 + 1/(3*(sqrt((27*y(x)*Derivative(y(x), (x, 2)) - 2)**2 - 4)/2 + 27*y(x)*Derivative(y(x), (x, 2))/2 - 1)**(1/3)) cannot be solved by the factorable group method