38.5.81 problem 68 (b)
Internal
problem
ID
[8429]
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
:
68
(b)
Date
solved
:
Tuesday, September 30, 2025 at 05:36:34 PM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {x \left (1-x \right )}{y \left (y-2\right )} \end{align*}
With initial conditions
\begin{align*}
y \left (0\right )&={\frac {3}{2}} \\
\end{align*}
✓ Maple. Time used: 0.726 (sec). Leaf size: 158
ode:=diff(y(x),x) = x*(1-x)/y(x)/(y(x)-2);
ic:=[y(0) = 3/2];
dsolve([ode,op(ic)],y(x), singsol=all);
\[
y = -\frac {\left (1+i \sqrt {3}\right ) \left (-44-32 x^{3}+48 x^{2}+4 \sqrt {64 x^{6}-192 x^{5}+144 x^{4}+176 x^{3}-264 x^{2}-135}\right )^{{2}/{3}}-16 i \sqrt {3}-8 \left (-44-32 x^{3}+48 x^{2}+4 \sqrt {64 x^{6}-192 x^{5}+144 x^{4}+176 x^{3}-264 x^{2}-135}\right )^{{1}/{3}}+16}{8 \left (-44-32 x^{3}+48 x^{2}+4 \sqrt {64 x^{6}-192 x^{5}+144 x^{4}+176 x^{3}-264 x^{2}-135}\right )^{{1}/{3}}}
\]
✓ Mathematica. Time used: 2.976 (sec). Leaf size: 242
ode=D[y[x],x]==(x*(1-x))/(y[x]*(y[x]-2));
ic={y[0]==3/2};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {-i 2^{2/3} \sqrt {3} \left (-8 x^3+12 x^2+\sqrt {64 x^6-192 x^5+144 x^4+176 x^3-264 x^2-135}-11\right )^{2/3}-2^{2/3} \left (-8 x^3+12 x^2+\sqrt {64 x^6-192 x^5+144 x^4+176 x^3-264 x^2-135}-11\right )^{2/3}+8 \sqrt [3]{-8 x^3+12 x^2+\sqrt {64 x^6-192 x^5+144 x^4+176 x^3-264 x^2-135}-11}+8 i \sqrt [3]{2} \sqrt {3}-8 \sqrt [3]{2}}{8 \sqrt [3]{-8 x^3+12 x^2+\sqrt {64 x^6-192 x^5+144 x^4+176 x^3-264 x^2-135}-11}} \end{align*}
✓ Sympy. Time used: 136.292 (sec). Leaf size: 204
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*(1 - x)/((y(x) - 2)*y(x)) + Derivative(y(x), x),0)
ics = {y(0): 3/2}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\sqrt [3]{\frac {27 x^{3}}{2} - \frac {81 x^{2}}{4} + \frac {\sqrt {729 x^{6} - 2187 x^{5} + \frac {6561 x^{4}}{4} + \frac {8019 x^{3}}{4} - \frac {24057 x^{2}}{8} - \frac {98415}{64}}}{2} + \frac {297}{16}}}{6} + \frac {\sqrt {3} i \sqrt [3]{\frac {27 x^{3}}{2} - \frac {81 x^{2}}{4} + \frac {\sqrt {729 x^{6} - 2187 x^{5} + \frac {6561 x^{4}}{4} + \frac {8019 x^{3}}{4} - \frac {24057 x^{2}}{8} - \frac {98415}{64}}}{2} + \frac {297}{16}}}{6} + 1 - \frac {6}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {27 x^{3}}{2} - \frac {81 x^{2}}{4} + \frac {\sqrt {729 x^{6} - 2187 x^{5} + \frac {6561 x^{4}}{4} + \frac {8019 x^{3}}{4} - \frac {24057 x^{2}}{8} - \frac {98415}{64}}}{2} + \frac {297}{16}}}
\]