47.4.11 problem 12

Internal problem ID [9785]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 06:41:13 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 }}^{3}-{y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 36

ode:=y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^3-diff(y(x),x)^2 = 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: 0.083 (sec). Leaf size: 47

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

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ][c_1-\log (K[2])]}dK[2]=x+c_2,y(x)\right ] \]

✗ 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)*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