89.8.14 problem 16

Internal problem ID [24474]
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 : 16
Date solved : Thursday, October 02, 2025 at 10:40:33 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} x +3 y-4+\left (x +4 y-5\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 33

ode:=x+3*y(x)-4+(x+4*y(x)-5)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-2 x +6\right ) \operatorname {LambertW}\left (\frac {c_1 \left (x -1\right )}{2}\right )-x +1}{4 \operatorname {LambertW}\left (\frac {c_1 \left (x -1\right )}{2}\right )} \]

✓ Mathematica. Time used: 0.766 (sec). Leaf size: 176

ode=(x+3*y[x]-4)+( x+4*y[x]-5 )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {(-1)^{2/3} \left (-2 x \log \left (\frac {6 (-2)^{2/3} (2 y(x)+x-3)}{4 y(x)+x-5}\right )+2 (x-3) \log \left (-\frac {3 (-2)^{2/3} (x-1)}{4 y(x)+x-5}\right )+6 \log \left (\frac {6 (-2)^{2/3} (2 y(x)+x-3)}{4 y(x)+x-5}\right )+4 y(x) \left (\log \left (-\frac {3 (-2)^{2/3} (x-1)}{4 y(x)+x-5}\right )-\log \left (\frac {6 (-2)^{2/3} (2 y(x)+x-3)}{4 y(x)+x-5}\right )+1\right )+x-5\right )}{9 \sqrt [3]{2} (2 y(x)+x-3)}=\frac {1}{9} (-2)^{2/3} \log (x-1)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.780 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x + (x + 4*y(x) - 5)*Derivative(y(x), x) + 3*y(x) - 4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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