23.2.226 problem 232

Internal problem ID [5581]
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 : 232
Date solved : Tuesday, September 30, 2025 at 01:00:29 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.241 (sec). Leaf size: 83

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

\begin{align*} y &= -\frac {\sqrt {4 x^{2}-2 a}}{2} \\ y &= \frac {\sqrt {4 x^{2}-2 a}}{2} \\ y &= -\frac {\sqrt {-8 c_1^{2}+16 c_1 x -4 x^{2}-2 a}}{2} \\ y &= \frac {\sqrt {-8 c_1^{2}+16 c_1 x -4 x^{2}-2 a}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.406 (sec). Leaf size: 63

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a - 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