23.2.200 problem 205

Internal problem ID [5555]
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 : 205
Date solved : Tuesday, September 30, 2025 at 12:53:11 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 54

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

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

✓ Sympy. Time used: 0.373 (sec). Leaf size: 29

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

\[ \left [ y{\left (x \right )} = - \sqrt {C_{1} - x^{2}}, \ y{\left (x \right )} = \sqrt {C_{1} - x^{2}}, \ y{\left (x \right )} = C_{1} e^{x}\right ] \]