23.2.74 problem 76

Internal problem ID [5429]
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 : 76
Date solved : Tuesday, September 30, 2025 at 12:42:06 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 34

ode=(D[y[x],x])^2-(x-y[x])*y[x]*D[y[x],x]-x*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 0 \end{align*}

✓ Sympy. Time used: 0.225 (sec). Leaf size: 17

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

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