89.8.8 problem 10

Internal problem ID [24468]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 66
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:39:56 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 2.14 (sec). Leaf size: 37

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

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

✓ Sympy. Time used: 1.919 (sec). Leaf size: 104

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

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