83.7.1 problem 1

Internal problem ID [21949]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. VIII at page 53
Problem number : 1
Date solved : Thursday, October 02, 2025 at 08:19:04 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 2-x -y+\left (x +y+3\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 21

ode:=2-x-y(x)+(3+x+y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -x +\frac {5 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{\frac {1}{5}+\frac {4 x}{5}}}{5}\right )}{2}-\frac {1}{2} \]

✓ Mathematica. Time used: 2.38 (sec). Leaf size: 41

ode=(2-x-y[x])+(3+x+y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (5 W\left (-e^{\frac {4 x}{5}-1+c_1}\right )-2 x-1\right )\\ y(x)&\to -x-\frac {1}{2} \end{align*}

✓ Sympy. Time used: 8.056 (sec). Leaf size: 243

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + (x + y(x) + 3)*Derivative(y(x), x) - y(x) + 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - x + \frac {5 W\left (- \frac {\sqrt [5]{C_{1} e^{4 x}} e^{\frac {1}{5}}}{10}\right )}{2} - \frac {1}{2}, \ y{\left (x \right )} = - x + \frac {5 W\left (- \frac {\sqrt [5]{C_{1} e^{4 x}} \left (-1 + \sqrt {5} + \sqrt {2} i \sqrt {\sqrt {5} + 5}\right ) e^{\frac {1}{5}}}{40}\right )}{2} - \frac {1}{2}, \ y{\left (x \right )} = - x + \frac {5 W\left (\frac {\sqrt [5]{C_{1} e^{4 x}} \left (1 + \sqrt {5} - \sqrt {2} i \sqrt {5 - \sqrt {5}}\right ) e^{\frac {1}{5}}}{40}\right )}{2} - \frac {1}{2}, \ y{\left (x \right )} = - x + \frac {5 W\left (\frac {\sqrt [5]{C_{1} e^{4 x}} \left (1 + \sqrt {5} + \sqrt {2} i \sqrt {5 - \sqrt {5}}\right ) e^{\frac {1}{5}}}{40}\right )}{2} - \frac {1}{2}, \ y{\left (x \right )} = - x + \frac {5 W\left (\frac {\sqrt [5]{C_{1} e^{4 x}} \left (- \sqrt {5} + 1 + \sqrt {2} i \sqrt {\sqrt {5} + 5}\right ) e^{\frac {1}{5}}}{40}\right )}{2} - \frac {1}{2}\right ] \]