89.10.19 problem 19

Internal problem ID [24512]
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 : 19
Date solved : Thursday, October 02, 2025 at 10:44:11 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 2 x +4 y-1-\left (x +2 y-3\right ) y^{\prime }&=0 \end{align*}

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

ode:=2*x+4*y(x)-1-(x+2*y(x)-3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 2.358 (sec). Leaf size: 43

ode=( 2*x+4*y[x]-1 )- ( x+2*y[x]-3 )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.007 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x - (x + 2*y(x) - 3)*Derivative(y(x), x) + 4*y(x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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