38.5.19 problem 19

Internal problem ID [8367]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 05:30:12 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {x y+3 x -y-3}{x y-2 x +4 y-8} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 23

ode:=diff(y(x),x) = (x*y(x)+3*x-y(x)-3)/(x*y(x)-2*x+4*y(x)-8); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -5 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {x}{5}-\frac {3}{5}-\frac {c_1}{5}} \left (x +4\right )}{5}\right )-3 \]

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 52

ode=D[y[x],x]==(x*y[x]+3*x-y[x]-3)/(x*y[x]-2*x+4*y[x]-8); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]-2}{K[1]+3}dK[1]\&\right ]\left [\int _1^x\frac {K[2]-1}{K[2]+4}dK[2]+c_1\right ]\\ y(x)&\to -3 \end{align*}

✓ Sympy. Time used: 6.568 (sec). Leaf size: 342

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x*y(x) + 3*x - y(x) - 3)/(x*y(x) - 2*x + 4*y(x) - 8),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (- x - 4\right )}{5 e^{\frac {3}{5}}}\right ) - 3, \ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (- \sqrt {5} x + x - \sqrt {2} i x \sqrt {\sqrt {5} + 5} - 4 \sqrt {5} + 4 - 4 \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{20 e^{\frac {3}{5}}}\right ) - 3, \ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (x + \sqrt {5} x - \sqrt {2} i x \sqrt {5 - \sqrt {5}} + 4 + 4 \sqrt {5} - 4 \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{20 e^{\frac {3}{5}}}\right ) - 3, \ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (x + \sqrt {5} x + \sqrt {2} i x \sqrt {5 - \sqrt {5}} + 4 + 4 \sqrt {5} + 4 \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{20 e^{\frac {3}{5}}}\right ) - 3, \ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (- \sqrt {5} x + x + \sqrt {2} i x \sqrt {\sqrt {5} + 5} - 4 \sqrt {5} + 4 + 4 \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{20 e^{\frac {3}{5}}}\right ) - 3\right ] \]