49.21.3 problem 1(c)
Internal
problem
ID
[7733]
Book
:
An
introduction
to
Ordinary
Differential
Equations.
Earl
A.
Coddington.
Dover.
NY
1961
Section
:
Chapter
5.
Existence
and
uniqueness
of
solutions
to
first
order
equations.
Page
190
Problem
number
:
1(c)
Date
solved
:
Sunday, March 30, 2025 at 12:20:21 PM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {x^{2}+x}{y-y^{2}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 494
ode:=diff(y(x),x) = (x^2+x)/(y(x)-y(x)^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (1-4 x^{3}-6 x^{2}-12 c_1 +2 \sqrt {4 x^{6}+12 x^{5}+24 c_1 \,x^{3}+9 x^{4}+36 c_1 \,x^{2}-2 x^{3}+36 c_1^{2}-3 x^{2}-6 c_1}\right )^{{1}/{3}}}{2}+\frac {1}{2 \left (1-4 x^{3}-6 x^{2}-12 c_1 +2 \sqrt {4 x^{6}+12 x^{5}+24 c_1 \,x^{3}+9 x^{4}+36 c_1 \,x^{2}-2 x^{3}+36 c_1^{2}-3 x^{2}-6 c_1}\right )^{{1}/{3}}}+\frac {1}{2} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (1-4 x^{3}-6 x^{2}-12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{2}/{3}}-i \sqrt {3}-2 \left (1-4 x^{3}-6 x^{2}-12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{1}/{3}}+1}{4 \left (1-4 x^{3}-6 x^{2}-12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (1-4 x^{3}-6 x^{2}-12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{2}/{3}}-i \sqrt {3}+2 \left (1-4 x^{3}-6 x^{2}-12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{1}/{3}}-1}{4 \left (1-4 x^{3}-6 x^{2}-12 c_1 +2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_1 \right ) \left (2 x^{3}+3 x^{2}+6 c_1 -1\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.356 (sec). Leaf size: 346
ode=D[y[x],x]==(x+x^2)/(y[x]-y[x]^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{2} \left (\sqrt [3]{-4 x^3-6 x^2+\sqrt {-1+\left (-4 x^3-6 x^2+1+12 c_1\right ){}^2}+1+12 c_1}+\frac {1}{\sqrt [3]{-4 x^3-6 x^2+\sqrt {-1+\left (-4 x^3-6 x^2+1+12 c_1\right ){}^2}+1+12 c_1}}+1\right ) \\
y(x)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-4 x^3-6 x^2+\sqrt {-1+\left (-4 x^3-6 x^2+1+12 c_1\right ){}^2}+1+12 c_1}+\frac {-2-2 i \sqrt {3}}{\sqrt [3]{-4 x^3-6 x^2+\sqrt {-1+\left (-4 x^3-6 x^2+1+12 c_1\right ){}^2}+1+12 c_1}}+4\right ) \\
y(x)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-4 x^3-6 x^2+\sqrt {-1+\left (-4 x^3-6 x^2+1+12 c_1\right ){}^2}+1+12 c_1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{-4 x^3-6 x^2+\sqrt {-1+\left (-4 x^3-6 x^2+1+12 c_1\right ){}^2}+1+12 c_1}}+4\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-x**2 - x)/(-y(x)**2 + y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ZeroDivisionError : polynomial division