89.8.10 problem 12

Internal problem ID [24470]
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 : 12
Date solved : Thursday, October 02, 2025 at 10:40:04 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.054 (sec). Leaf size: 28

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

\[ y = \frac {-x \operatorname {LambertW}\left (-c_1 \left (x -2\right )\right )+x -2}{2 \operatorname {LambertW}\left (-c_1 \left (x -2\right )\right )} \]

✓ Mathematica. Time used: 2.203 (sec). Leaf size: 109

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

\[ \text {Solve}\left [\frac {2^{2/3} \left (x \log \left (\frac {x-2}{y(x)+x-1}\right )-x \log \left (\frac {2 y(x)+x}{y(x)+x-1}\right )+2 y(x) \left (\log \left (\frac {x-2}{y(x)+x-1}\right )-\log \left (\frac {2 y(x)+x}{y(x)+x-1}\right )+1\right )+2 x-2\right )}{9 (2 y(x)+x)}=\frac {1}{9} 2^{2/3} \log (x-2)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.641 (sec). Leaf size: 19

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

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