23.2.225 problem 231

Internal problem ID [5580]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 231
Date solved : Tuesday, September 30, 2025 at 01:00:27 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.494 (sec). Leaf size: 107

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

\begin{align*} y &= -x \\ y &= x \\ y &= \sqrt {-2 \sqrt {2}\, c_1 x -c_1^{2}-x^{2}} \\ y &= \sqrt {2 \sqrt {2}\, c_1 x -c_1^{2}-x^{2}} \\ y &= -\sqrt {-2 \sqrt {2}\, c_1 x -c_1^{2}-x^{2}} \\ y &= -\sqrt {2 \sqrt {2}\, c_1 x -c_1^{2}-x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.454 (sec). Leaf size: 171

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

\begin{align*} y(x)&\to -\frac {\sqrt {-2 x^2-4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-2 x^2-4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-2 x^2+4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-2 x^2+4 i e^{c_1} x+e^{2 c_1}}}{\sqrt {2}}\\ y(x)&\to -\sqrt {-x^2}\\ y(x)&\to \sqrt {-x^2} \end{align*}

✗ Sympy

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

Timed Out