71.1.20 problem 36 (page 40)

Internal problem ID [19196]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 36 (page 40)
Date solved : Thursday, October 02, 2025 at 03:43:01 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 19

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

\[ y = \frac {1}{\operatorname {RootOf}\left (-\textit {\_Z}^{2} x +{\mathrm e}^{\textit {\_Z}} c_1 +1\right )} \]

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 25

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

\[ \text {Solve}\left [x=y(x)^2+c_1 e^{\frac {1}{y(x)}-2} y(x)^2,y(x)\right ] \]

✓ Sympy. Time used: 0.729 (sec). Leaf size: 24

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

\[ C_{1} + \frac {x e^{- \frac {1}{y{\left (x \right )}}}}{y^{2}{\left (x \right )}} - e^{- \frac {1}{y{\left (x \right )}}} = 0 \]