23.4.152 problem 152

Internal problem ID [6454]
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 : 152
Date solved : Tuesday, September 30, 2025 at 02:57:22 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 40

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

\begin{align*} y &= 0 \\ y &= \frac {{\mathrm e}^{x}}{\sqrt {2 c_2 \,{\mathrm e}^{x}-2 c_1}} \\ y &= -\frac {{\mathrm e}^{x}}{\sqrt {2 c_2 \,{\mathrm e}^{x}-2 c_1}} \\ \end{align*}

✓ Mathematica. Time used: 7.625 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to \frac {c_2 e^{x+c_1}}{\sqrt {-1+2 e^{x+c_1}}}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(3)*sqrt((-y(x) + 4*Derivative(y(x), (x, 2)))*y(x))/6 - y(x