88.12.15 problem 17
Internal
problem
ID
[24071]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
2.
Differential
equations
of
first
order.
Miscellaneous
Exercises
at
page
55
Problem
number
:
17
Date
solved
:
Thursday, October 02, 2025 at 09:56:57 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational]
\begin{align*} y^{\prime }&=\frac {y}{y-y^{3}+2 x} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 407
ode:=diff(y(x),x) = y(x)/(y(x)-y(x)^3+2*x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-36 c_1 -108 x +8 c_1^{3}+12 \sqrt {-12 x \,c_1^{3}-3 c_1^{2}+54 c_1 x +81 x^{2}+12}\right )^{{1}/{3}}}{6}+\frac {\frac {2 c_1^{2}}{3}-2}{\left (-36 c_1 -108 x +8 c_1^{3}+12 \sqrt {-12 x \,c_1^{3}-3 c_1^{2}+54 c_1 x +81 x^{2}+12}\right )^{{1}/{3}}}+\frac {c_1}{3} \\
y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (-36 c_1 -108 x +8 c_1^{3}+12 \sqrt {81 x^{2}+\left (-12 c_1^{3}+54 c_1 \right ) x -3 c_1^{2}+12}\right )^{{2}/{3}}}{4}+c_1 \left (-36 c_1 -108 x +8 c_1^{3}+12 \sqrt {81 x^{2}+\left (-12 c_1^{3}+54 c_1 \right ) x -3 c_1^{2}+12}\right )^{{1}/{3}}+\left (i \sqrt {3}-1\right ) \left (c_1^{2}-3\right )}{3 \left (-36 c_1 -108 x +8 c_1^{3}+12 \sqrt {81 x^{2}+\left (-12 c_1^{3}+54 c_1 \right ) x -3 c_1^{2}+12}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-36 c_1 -108 x +8 c_1^{3}+12 \sqrt {81 x^{2}+\left (-12 c_1^{3}+54 c_1 \right ) x -3 c_1^{2}+12}\right )^{{2}/{3}}}{4}+c_1 \left (-36 c_1 -108 x +8 c_1^{3}+12 \sqrt {81 x^{2}+\left (-12 c_1^{3}+54 c_1 \right ) x -3 c_1^{2}+12}\right )^{{1}/{3}}+\left (-i \sqrt {3}-1\right ) \left (c_1^{2}-3\right )}{3 \left (-36 c_1 -108 x +8 c_1^{3}+12 \sqrt {81 x^{2}+\left (-12 c_1^{3}+54 c_1 \right ) x -3 c_1^{2}+12}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 10.728 (sec). Leaf size: 472
ode=D[y[x],{x,1}]==y[x]/(y[x]-y[x]^3+2*x);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {27 x^2-4 c_1{}^3 x+18 c_1 x+4-c_1{}^2}-27 x+2 c_1{}^3-9 c_1}+\frac {2 \sqrt [3]{2} \left (-3+c_1{}^2\right )}{\sqrt [3]{3 \sqrt {3} \sqrt {27 x^2-4 c_1{}^3 x+18 c_1 x+4-c_1{}^2}-27 x+2 c_1{}^3-9 c_1}}+2 c_1\right )\\ y(x)&\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{3 \sqrt {3} \sqrt {27 x^2-4 c_1{}^3 x+18 c_1 x+4-c_1{}^2}-27 x+2 c_1{}^3-9 c_1}-\frac {2 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) \left (-3+c_1{}^2\right )}{\sqrt [3]{3 \sqrt {3} \sqrt {27 x^2-4 c_1{}^3 x+18 c_1 x+4-c_1{}^2}-27 x+2 c_1{}^3-9 c_1}}+4 c_1\right )\\ y(x)&\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{3 \sqrt {3} \sqrt {27 x^2-4 c_1{}^3 x+18 c_1 x+4-c_1{}^2}-27 x+2 c_1{}^3-9 c_1}+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (-3+c_1{}^2\right )}{\sqrt [3]{3 \sqrt {3} \sqrt {27 x^2-4 c_1{}^3 x+18 c_1 x+4-c_1{}^2}-27 x+2 c_1{}^3-9 c_1}}+4 c_1\right )\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 177.684 (sec). Leaf size: 352
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - y(x)/(2*x - y(x)**3 + y(x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {C_{1}}{3} - \frac {C_{1}^{2} - 3}{3 \sqrt [3]{C_{1}^{3} - \frac {9 C_{1}}{2} + \frac {27 x}{2} + \frac {\sqrt {- 4 \left (C_{1}^{2} - 3\right )^{3} + \left (2 C_{1}^{3} - 9 C_{1} + 27 x\right )^{2}}}{2}}} - \frac {\sqrt [3]{C_{1}^{3} - \frac {9 C_{1}}{2} + \frac {27 x}{2} + \frac {\sqrt {- 4 \left (C_{1}^{2} - 3\right )^{3} + \left (2 C_{1}^{3} - 9 C_{1} + 27 x\right )^{2}}}{2}}}{3}, \ y{\left (x \right )} = - \frac {C_{1}}{3} - \frac {2 \left (C_{1}^{2} - 3\right )}{3 \left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{3} - \frac {9 C_{1}}{2} + \frac {27 x}{2} + \frac {\sqrt {- 4 \left (C_{1}^{2} - 3\right )^{3} + \left (2 C_{1}^{3} - 9 C_{1} + 27 x\right )^{2}}}{2}}} - \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{3} - \frac {9 C_{1}}{2} + \frac {27 x}{2} + \frac {\sqrt {- 4 \left (C_{1}^{2} - 3\right )^{3} + \left (2 C_{1}^{3} - 9 C_{1} + 27 x\right )^{2}}}{2}}}{6}, \ y{\left (x \right )} = - \frac {C_{1}}{3} - \frac {2 \left (C_{1}^{2} - 3\right )}{3 \left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{3} - \frac {9 C_{1}}{2} + \frac {27 x}{2} + \frac {\sqrt {- 4 \left (C_{1}^{2} - 3\right )^{3} + \left (2 C_{1}^{3} - 9 C_{1} + 27 x\right )^{2}}}{2}}} - \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{3} - \frac {9 C_{1}}{2} + \frac {27 x}{2} + \frac {\sqrt {- 4 \left (C_{1}^{2} - 3\right )^{3} + \left (2 C_{1}^{3} - 9 C_{1} + 27 x\right )^{2}}}{2}}}{6}\right ]
\]