68.4.65 problem 61 (b)
Internal
problem
ID
[17245]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Exercises
2.2,
page
39
Problem
number
:
61
(b)
Date
solved
:
Thursday, October 02, 2025 at 01:59:51 PM
CAS
classification
:
[_quadrature]
\begin{align*} y^{\prime }&=\left (3 y+1\right )^{4} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 116
ode:=diff(y(x),x) = (3*y(x)+1)^4;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {3^{{1}/{3}} \left (-\left (c_1 +x \right )^{2}\right )^{{1}/{3}}-3 c_1 -3 x}{9 c_1 +9 x} \\
y &= \frac {\left (-i 3^{{5}/{6}}-3^{{1}/{3}}\right ) \left (-\left (c_1 +x \right )^{2}\right )^{{1}/{3}}-6 x -6 c_1}{18 c_1 +18 x} \\
y &= \frac {\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) \left (-\left (c_1 +x \right )^{2}\right )^{{1}/{3}}-6 x -6 c_1}{18 c_1 +18 x} \\
\end{align*}
✓ Mathematica. Time used: 0.559 (sec). Leaf size: 144
ode=D[y[x],x]==(3*y[x]+1)^4;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {3 x+\sqrt [3]{3} \sqrt [3]{(x+c_1){}^2}+3 c_1}{9 (x+c_1)}\\ y(x)&\to \frac {-6 x+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \sqrt [3]{(x+c_1){}^2}-6 c_1}{18 (x+c_1)}\\ y(x)&\to \frac {-6 x+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \sqrt [3]{(x+c_1){}^2}-6 c_1}{18 (x+c_1)}\\ y(x)&\to -\frac {1}{3} \end{align*}
✓ Sympy. Time used: 1.623 (sec). Leaf size: 173
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-(3*y(x) + 1)**4 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt [3]{3} \sqrt [3]{\frac {9 C_{1}}{- C_{1} + x} - \frac {9 x}{- C_{1} + x} + 9 - \frac {1}{- C_{1} + x}}}{18} - \frac {3^{\frac {5}{6}} i \sqrt [3]{\frac {9 C_{1}}{- C_{1} + x} - \frac {9 x}{- C_{1} + x} + 9 - \frac {1}{- C_{1} + x}}}{18} - \frac {1}{3}, \ y{\left (x \right )} = - \frac {\sqrt [3]{3} \sqrt [3]{\frac {9 C_{1}}{- C_{1} + x} - \frac {9 x}{- C_{1} + x} + 9 - \frac {1}{- C_{1} + x}}}{18} + \frac {3^{\frac {5}{6}} i \sqrt [3]{\frac {9 C_{1}}{- C_{1} + x} - \frac {9 x}{- C_{1} + x} + 9 - \frac {1}{- C_{1} + x}}}{18} - \frac {1}{3}, \ y{\left (x \right )} = \frac {\sqrt [3]{3} \sqrt [3]{\frac {9 C_{1}}{- C_{1} + x} - \frac {9 x}{- C_{1} + x} + 9 - \frac {1}{- C_{1} + x}}}{9} - \frac {1}{3}\right ]
\]