89.8.13 problem 15

Internal problem ID [24473]
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 : 15
Date solved : Thursday, October 02, 2025 at 10:40:27 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} 9 x -4 y+4-\left (1+2 x -y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 24

ode:=9*x-4*y(x)+4-(2*x-y(x)+1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 1+\frac {x \left (3 \operatorname {LambertW}\left (-c_1 x \right )+1\right )}{\operatorname {LambertW}\left (-c_1 x \right )} \]

✓ Mathematica. Time used: 0.448 (sec). Leaf size: 182

ode=(9*x-4*y[x]+4)-(2*x-y[x]+1 )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {(-2)^{2/3} \left (-3 x \log \left (\frac {3 (-2)^{2/3} (-y(x)+3 x+1)}{-y(x)+2 x+1}\right )+(3 x+1) \log \left (-\frac {3 (-2)^{2/3} x}{-y(x)+2 x+1}\right )-\log \left (\frac {3 (-2)^{2/3} (-y(x)+3 x+1)}{-y(x)+2 x+1}\right )+y(x) \left (-\log \left (-\frac {3 (-2)^{2/3} x}{-y(x)+2 x+1}\right )+\log \left (\frac {3 (-2)^{2/3} (-y(x)+3 x+1)}{-y(x)+2 x+1}\right )-1\right )+2 x+1\right )}{-9 y(x)+27 x+9}=\frac {1}{9} (-2)^{2/3} \log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.714 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x - (2*x - y(x) + 1)*Derivative(y(x), x) - 4*y(x) + 4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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