23.1.606 problem 600
Internal
problem
ID
[5213]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
600
Date
solved
:
Tuesday, September 30, 2025 at 11:55:55 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (x -y^{2}\right ) y^{\prime }&=x^{2}-y \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 316
ode:=(x-y(x)^2)*diff(y(x),x) = x^2-y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 x}{2 \left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (-\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 x \right ) \sqrt {3}-\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x}{4 \left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x \right ) \sqrt {3}-\left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x}{4 \left (-4 x^{3}+12 c_1 +4 \sqrt {x^{6}+\left (-6 c_1 -4\right ) x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 2.655 (sec). Leaf size: 326
ode=(x-y[x]^2)*D[y[x],x]==x^2-y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {2 x+\sqrt [3]{2} \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}}\\ y(x)&\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}+\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x}{4 \sqrt [3]{x^3+\sqrt {x^6+(-4+6 c_1) x^3+9 c_1{}^2}+3 c_1}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 + (x - y(x)**2)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out