81.15.2 problem 19-2

Internal problem ID [21710]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 19. Change of variables. Page 483
Problem number : 19-2
Date solved : Thursday, October 02, 2025 at 08:00:32 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 20

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

\[ y = -x +\frac {\operatorname {LambertW}\left (c_1 \,{\mathrm e}^{3+4 x}\right )}{2}-\frac {3}{2} \]

✓ Mathematica. Time used: 2.342 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to \frac {1}{2} \left (W\left (-e^{4 x-1+c_1}\right )-2 x-3\right )\\ y(x)&\to -x-\frac {3}{2} \end{align*}

✓ Sympy. Time used: 0.593 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = - x + \frac {W\left (C_{1} e^{4 x + 3}\right )}{2} - \frac {3}{2} \]