43.22.6 problem 1(f)

Internal problem ID [9038]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 5. Existence and uniqueness of solutions to first order equations. Page 198
Problem number : 1(f)
Date solved : Tuesday, September 30, 2025 at 06:02:20 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: 0.017 (sec). Leaf size: 49

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: 0.271 (sec). Leaf size: 86

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

✓ Sympy. Time used: 0.711 (sec). Leaf size: 27

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)
 

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