69.7.16 problem 191

Internal problem ID [18096]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 191
Date solved : Thursday, October 02, 2025 at 02:42:01 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

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

\begin{align*} y &= \sqrt {\left (\ln \left (x \right )+c_1 \right ) x} \\ y &= -\sqrt {\left (\ln \left (x \right )+c_1 \right ) x} \\ \end{align*}

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 40

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

\begin{align*} y(x)&\to -\sqrt {x} \sqrt {\log (x)+c_1}\\ y(x)&\to \sqrt {x} \sqrt {\log (x)+c_1} \end{align*}

✓ Sympy. Time used: 0.270 (sec). Leaf size: 26

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

\[ \left [ y{\left (x \right )} = - \sqrt {x \left (C_{1} + \log {\left (x \right )}\right )}, \ y{\left (x \right )} = \sqrt {x \left (C_{1} + \log {\left (x \right )}\right )}\right ] \]