44.5.19 problem 19

Internal problem ID [7081]
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 : Monday, January 27, 2025 at 02:47:22 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 23

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

\[ y = -5 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {x}{5}-\frac {3}{5}-\frac {c_{1}}{5}} \left (x +4\right )}{5}\right )-3 \]

✓ Solution by Mathematica

Time used: 60.087 (sec). Leaf size: 216

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

\begin{align*} y(x)\to -3-5 W\left (\frac {\sqrt [5]{(x+4)^5 \left (-e^{-x-c_1}\right )}}{5 e^{3/5}}\right ) \\ y(x)\to -3-5 W\left (-\frac {\sqrt [5]{-1} \sqrt [5]{(x+4)^5 \left (-e^{-x-c_1}\right )}}{5 e^{3/5}}\right ) \\ y(x)\to -3-5 W\left (\frac {(-1)^{2/5} \sqrt [5]{(x+4)^5 \left (-e^{-x-c_1}\right )}}{5 e^{3/5}}\right ) \\ y(x)\to -3-5 W\left (-\frac {(-1)^{3/5} \sqrt [5]{(x+4)^5 \left (-e^{-x-c_1}\right )}}{5 e^{3/5}}\right ) \\ y(x)\to -3-5 W\left (\frac {(-1)^{4/5} \sqrt [5]{(x+4)^5 \left (-e^{-x-c_1}\right )}}{5 e^{3/5}}\right ) \\ \end{align*}