23.4.217 problem 217

Internal problem ID [6519]
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 : 217
Date solved : Tuesday, September 30, 2025 at 03:02:37 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 25

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

\begin{align*} y &= 0 \\ y &= c_1 \sqrt {x}\, c_2 +c_1^{2} x +\frac {c_2^{2}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 18

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*y(x)*Derivative(y(x), (x, 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)) + y