23.2.287 problem 306

Internal problem ID [5642]
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 : 306
Date solved : Tuesday, September 30, 2025 at 01:24:01 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}-\left (x^{2}+x y+y^{2}\right ) {y^{\prime }}^{2}+x y \left (x^{2}+x y+y^{2}\right ) y^{\prime }-x^{3} y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 32

ode:=diff(y(x),x)^3-(x^2+x*y(x)+y(x)^2)*diff(y(x),x)^2+x*y(x)*(x^2+x*y(x)+y(x)^2)*diff(y(x),x)-x^3*y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{3}}{3}+c_1 \\ y &= \frac {1}{-x +c_1} \\ y &= c_1 \,{\mathrm e}^{\frac {x^{2}}{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 48

ode=(D[y[x],x])^3 -(x^2+x y[x]+ y[x]^2) (D[y[x],x])^2 +x y[x](x^2 +x y[x]+ y[x]^2) D[y[x],x]-x^3 y[x]^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{x+c_1}\\ y(x)&\to c_1 e^{\frac {x^2}{2}}\\ y(x)&\to \frac {x^3}{3}+c_1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.264 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*y(x)**3 + x*(x**2 + x*y(x) + y(x)**2)*y(x)*Derivative(y(x), x) - (x**2 + x*y(x) + y(x)**2)*Derivative(y(x), x)**2 + Derivative(y(x), x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} + \frac {x^{3}}{3}, \ y{\left (x \right )} = - \frac {1}{C_{1} + x}, \ y{\left (x \right )} = C_{1} e^{\frac {x^{2}}{2}}\right ] \]