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
:
Sunday, March 30, 2025 at 11:38:49 AM
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: 60.087 (sec). Leaf size: 216
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 -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*}
✓ Sympy. Time used: 9.962 (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)
\[
\left [ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (- x - 4\right )}{5 e^{\frac {3}{5}}}\right ) - 3, \ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (- \sqrt {5} x + x - \sqrt {2} i x \sqrt {\sqrt {5} + 5} - 4 \sqrt {5} + 4 - 4 \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{20 e^{\frac {3}{5}}}\right ) - 3, \ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (x + \sqrt {5} x - \sqrt {2} i x \sqrt {5 - \sqrt {5}} + 4 + 4 \sqrt {5} - 4 \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{20 e^{\frac {3}{5}}}\right ) - 3, \ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (x + \sqrt {5} x + \sqrt {2} i x \sqrt {5 - \sqrt {5}} + 4 + 4 \sqrt {5} + 4 \sqrt {2} i \sqrt {5 - \sqrt {5}}\right )}{20 e^{\frac {3}{5}}}\right ) - 3, \ y{\left (x \right )} = - 5 W\left (\frac {\sqrt [5]{C_{1} e^{- x}} \left (- \sqrt {5} x + x + \sqrt {2} i x \sqrt {\sqrt {5} + 5} - 4 \sqrt {5} + 4 + 4 \sqrt {2} i \sqrt {\sqrt {5} + 5}\right )}{20 e^{\frac {3}{5}}}\right ) - 3\right ]
\]