89.10.28 problem 29

Internal problem ID [24521]
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 77
Problem number : 29
Date solved : Thursday, October 02, 2025 at 10:45:12 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 3 x -3 y-2-\left (x -y+1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 19

ode:=3*x-3*y(x)-2-(x-y(x)+1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x -\frac {5 \operatorname {LambertW}\left (-\frac {c_1 \,{\mathrm e}^{-\frac {3}{5}-\frac {4 x}{5}}}{5}\right )}{2}-\frac {3}{2} \]

✓ Mathematica. Time used: 2.207 (sec). Leaf size: 37

ode=( 3*x-3*y[x]-2)-( x-y[x]+1 )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {5}{2} \left (1+W\left (-e^{-\frac {4 x}{5}-1+c_1}\right )\right )+x+1\\ y(x)&\to x-\frac {3}{2} \end{align*}

✓ Sympy. Time used: 8.544 (sec). Leaf size: 245

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x - (x - y(x) + 1)*Derivative(y(x), x) - 3*y(x) - 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x - \frac {5 W\left (\frac {\sqrt [5]{C_{1} e^{- 4 x}}}{5 e^{\frac {3}{5}}}\right )}{2} - \frac {3}{2}, \ y{\left (x \right )} = x - \frac {5 W\left (\frac {\sqrt [5]{C_{1} e^{- 4 x}} \left (-1 + \sqrt {5} + \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{20 e^{\frac {3}{5}}}\right )}{2} - \frac {3}{2}, \ y{\left (x \right )} = x - \frac {5 W\left (- \frac {\sqrt [5]{C_{1} e^{- 4 x}} \left (1 + \sqrt {5} - \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{20 e^{\frac {3}{5}}}\right )}{2} - \frac {3}{2}, \ y{\left (x \right )} = x - \frac {5 W\left (- \frac {\sqrt [5]{C_{1} e^{- 4 x}} \left (1 + \sqrt {5} + \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{20 e^{\frac {3}{5}}}\right )}{2} - \frac {3}{2}, \ y{\left (x \right )} = x - \frac {5 W\left (- \frac {\sqrt [5]{C_{1} e^{- 4 x}} \left (- \sqrt {5} + 1 + \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{20 e^{\frac {3}{5}}}\right )}{2} - \frac {3}{2}\right ] \]