89.6.18 problem 18

Internal problem ID [24401]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 18
Date solved : Thursday, October 02, 2025 at 10:25:04 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 51

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

\begin{align*} y &= \frac {-c_1 x -\sqrt {2 x^{2} c_1^{2}+1}}{c_1} \\ y &= \frac {-c_1 x +\sqrt {2 x^{2} c_1^{2}+1}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.254 (sec). Leaf size: 94

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

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

✓ Sympy. Time used: 0.693 (sec). Leaf size: 29

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

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