82.8.18 problem 36-19

Internal problem ID [21889]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-19
Date solved : Thursday, October 02, 2025 at 08:05:46 PM
CAS classification : [_quadrature]

\begin{align*} x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+1&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 32

ode:=x*y(x)*diff(y(x),x)^2+(x+y(x))*diff(y(x),x)+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\ln \left (x \right )+c_1 \\ y &= \sqrt {-2 x +c_1} \\ y &= -\sqrt {-2 x +c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 53

ode=x*y[x]*D[y[x],x]^2+(x+y[x])*D[y[x],x]+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.445 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)*Derivative(y(x), x)**2 + (x + y(x))*Derivative(y(x), x) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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