38.4.41 problem 23
Internal
problem
ID
[8340]
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
2.1
Solution
curves
without
a
solution.
Exercises
2.1
at
page
44
Problem
number
:
23
Date
solved
:
Tuesday, September 30, 2025 at 05:29:22 PM
CAS
classification
:
[_quadrature]
\begin{align*} y^{\prime }&=\left (y-2\right )^{4} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 116
ode:=diff(y(x),x) = (y(x)-2)^4;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {3^{{2}/{3}} \left (-\left (c_1 +x \right )^{2}\right )^{{1}/{3}}+6 c_1 +6 x}{3 c_1 +3 x} \\
y &= \frac {\left (-3 i 3^{{1}/{6}}-3^{{2}/{3}}\right ) \left (-\left (c_1 +x \right )^{2}\right )^{{1}/{3}}+12 x +12 c_1}{6 c_1 +6 x} \\
y &= \frac {\left (3 i 3^{{1}/{6}}-3^{{2}/{3}}\right ) \left (-\left (c_1 +x \right )^{2}\right )^{{1}/{3}}+12 x +12 c_1}{6 c_1 +6 x} \\
\end{align*}
✓ Mathematica. Time used: 0.573 (sec). Leaf size: 133
ode=D[y[x],x]==(y[x]-2)^4;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{3} \left (6-\frac {3^{2/3} \sqrt [3]{(x+c_1){}^2}}{x+c_1}\right )\\ y(x)&\to \frac {12 x+\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) \sqrt [3]{(x+c_1){}^2}+12 c_1}{6 (x+c_1)}\\ y(x)&\to \frac {12 x+\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) \sqrt [3]{(x+c_1){}^2}+12 c_1}{6 (x+c_1)}\\ y(x)&\to 2 \end{align*}
✓ Sympy. Time used: 1.828 (sec). Leaf size: 167
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-(y(x) - 2)**4 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt [3]{- \frac {8 C_{1}}{- C_{1} + x} + \frac {8 x}{- C_{1} + x} - 8 - \frac {1}{3 \left (- C_{1} + x\right )}}}{2} - \frac {\sqrt {3} i \sqrt [3]{- \frac {8 C_{1}}{- C_{1} + x} + \frac {8 x}{- C_{1} + x} - 8 - \frac {1}{3 \left (- C_{1} + x\right )}}}{2} + 2, \ y{\left (x \right )} = - \frac {\sqrt [3]{- \frac {8 C_{1}}{- C_{1} + x} + \frac {8 x}{- C_{1} + x} - 8 - \frac {1}{3 \left (- C_{1} + x\right )}}}{2} + \frac {\sqrt {3} i \sqrt [3]{- \frac {8 C_{1}}{- C_{1} + x} + \frac {8 x}{- C_{1} + x} - 8 - \frac {1}{3 \left (- C_{1} + x\right )}}}{2} + 2, \ y{\left (x \right )} = \sqrt [3]{- \frac {8 C_{1}}{- C_{1} + x} + \frac {8 x}{- C_{1} + x} - 8 - \frac {1}{3 \left (- C_{1} + x\right )}} + 2\right ]
\]