85.10.1 problem 1
Internal
problem
ID
[22488]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
two.
First
order
and
simple
higher
order
ordinary
differential
equations.
B
Exercises
at
page
37
Problem
number
:
1
Date
solved
:
Thursday, October 02, 2025 at 08:41:05 PM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {\left (y-1\right ) \left (3+y\right )}{\left (-2+y\right ) \left (x +3\right )} \end{align*}
✓ Maple. Time used: 0.323 (sec). Leaf size: 155
ode:=diff(y(x),x) = (y(x)-1)*(3+y(x))/(y(x)-2)/(x+3);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {3+c_1 \left (x +3\right )^{4} {\operatorname {RootOf}\left (1+\left (4 c_1 \,x^{4}+48 c_1 \,x^{3}+216 c_1 \,x^{2}+432 c_1 x +324 c_1 \right ) \textit {\_Z}^{20}+\left (-c_1 \,x^{4}-12 c_1 \,x^{3}-54 c_1 \,x^{2}-108 c_1 x -81 c_1 \right ) \textit {\_Z}^{16}\right )}^{16}}{-1+c_1 \left (x +3\right )^{4} {\operatorname {RootOf}\left (1+\left (4 c_1 \,x^{4}+48 c_1 \,x^{3}+216 c_1 \,x^{2}+432 c_1 x +324 c_1 \right ) \textit {\_Z}^{20}+\left (-c_1 \,x^{4}-12 c_1 \,x^{3}-54 c_1 \,x^{2}-108 c_1 x -81 c_1 \right ) \textit {\_Z}^{16}\right )}^{16}}
\]
✓ Mathematica. Time used: 5.558 (sec). Leaf size: 686
ode=D[y[x],{x,1}]==( (y[x]-1)*(x-1)*(y[x]+3) )/( (x-1)* (y[x]-2)* (x+3));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {Root}\left [\text {$\#$1}^5+15 \text {$\#$1}^4+90 \text {$\#$1}^3+270 \text {$\#$1}^2+\text {$\#$1} \left (-e^{4 c_1} x^4-12 e^{4 c_1} x^3-54 e^{4 c_1} x^2-108 e^{4 c_1} x+405-81 e^{4 c_1}\right )+e^{4 c_1} x^4+12 e^{4 c_1} x^3+54 e^{4 c_1} x^2+108 e^{4 c_1} x+243+81 e^{4 c_1}\&,1\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^5+15 \text {$\#$1}^4+90 \text {$\#$1}^3+270 \text {$\#$1}^2+\text {$\#$1} \left (-e^{4 c_1} x^4-12 e^{4 c_1} x^3-54 e^{4 c_1} x^2-108 e^{4 c_1} x+405-81 e^{4 c_1}\right )+e^{4 c_1} x^4+12 e^{4 c_1} x^3+54 e^{4 c_1} x^2+108 e^{4 c_1} x+243+81 e^{4 c_1}\&,2\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^5+15 \text {$\#$1}^4+90 \text {$\#$1}^3+270 \text {$\#$1}^2+\text {$\#$1} \left (-e^{4 c_1} x^4-12 e^{4 c_1} x^3-54 e^{4 c_1} x^2-108 e^{4 c_1} x+405-81 e^{4 c_1}\right )+e^{4 c_1} x^4+12 e^{4 c_1} x^3+54 e^{4 c_1} x^2+108 e^{4 c_1} x+243+81 e^{4 c_1}\&,3\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^5+15 \text {$\#$1}^4+90 \text {$\#$1}^3+270 \text {$\#$1}^2+\text {$\#$1} \left (-e^{4 c_1} x^4-12 e^{4 c_1} x^3-54 e^{4 c_1} x^2-108 e^{4 c_1} x+405-81 e^{4 c_1}\right )+e^{4 c_1} x^4+12 e^{4 c_1} x^3+54 e^{4 c_1} x^2+108 e^{4 c_1} x+243+81 e^{4 c_1}\&,4\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^5+15 \text {$\#$1}^4+90 \text {$\#$1}^3+270 \text {$\#$1}^2+\text {$\#$1} \left (-e^{4 c_1} x^4-12 e^{4 c_1} x^3-54 e^{4 c_1} x^2-108 e^{4 c_1} x+405-81 e^{4 c_1}\right )+e^{4 c_1} x^4+12 e^{4 c_1} x^3+54 e^{4 c_1} x^2+108 e^{4 c_1} x+243+81 e^{4 c_1}\&,5\right ]\\ y(x)&\to -3\\ y(x)&\to 1 \end{align*}
✓ Sympy. Time used: 0.194 (sec). Leaf size: 24
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (y(x) - 1)*(y(x) + 3)/((x + 3)*(y(x) - 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
- \log {\left (x + 3 \right )} - \frac {\log {\left (y{\left (x \right )} - 1 \right )}}{4} + \frac {5 \log {\left (y{\left (x \right )} + 3 \right )}}{4} = C_{1}
\]