89.8.11 problem 13

Internal problem ID [24471]
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 : 13
Date solved : Thursday, October 02, 2025 at 10:40:13 PM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 33

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

\[ y = 1+\frac {\tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x -1\right )+2 c_1 \right )\right ) \left (1-x \right )}{3} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 64

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

\[ \text {Solve}\left [2 \arctan \left (\frac {-3 y(x)+x+2}{-3 y(x)-x+4}\right )+\log \left (\frac {x^2+9 y(x)^2-18 y(x)-2 x+10}{2 (x-1)^2}\right )+2 \log (x-1)+c_1=0,y(x)\right ] \]

✓ Sympy. Time used: 2.342 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x + (3*x + 9*y(x) - 12)*Derivative(y(x), x) - 3*y(x) + 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \log {\left (x - 1 \right )} = C_{1} - \log {\left (\sqrt {\frac {1}{9} + \frac {\left (y{\left (x \right )} - 1\right )^{2}}{\left (x - 1\right )^{2}}} \right )} - \operatorname {atan}{\left (\frac {3 \left (y{\left (x \right )} - 1\right )}{x - 1} \right )} \]