23.4.208 problem 208

Internal problem ID [6510]
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 : 208
Date solved : Tuesday, September 30, 2025 at 03:02:31 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 67

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

\begin{align*} y &= 0 \\ y &= \left (3 c_1 \ln \left (x \right )+3 c_2 \right )^{{1}/{3}} \\ y &= -\frac {\left (3 c_1 \ln \left (x \right )+3 c_2 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (3 c_1 \ln \left (x \right )+3 c_2 \right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.175 (sec). Leaf size: 19

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

\begin{align*} y(x)&\to c_2 \sqrt [3]{3 \log (x)+c_1} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((-8*x**2*Derivative(y(x), (x, 2)) +