23.1.621 problem 615
Internal
problem
ID
[5228]
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
:
615
Date
solved
:
Tuesday, September 30, 2025 at 11:56:35 AM
CAS
classification
:
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\begin{align*} \left (x^{3}+2 y-y^{2}\right ) y^{\prime }+3 x^{2} y&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 442
ode:=(x^3+2*y(x)-y(x)^2)*diff(y(x),x)+3*x^2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{1}/{3}}}{2}+\frac {2 x^{3}+2}{\left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{1}/{3}}}+1 \\
y &= 1-\frac {\left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{1}/{3}}}{4}+\frac {-x^{3}-1}{\left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{1}/{3}}}+\frac {i \left (x^{3}-\frac {\left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{2}/{3}}}{4}+1\right ) \sqrt {3}}{\left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{2}/{3}}}{4}-1+\left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{1}/{3}}-i \left (x^{3}+1\right ) \sqrt {3}-x^{3}}{\left (12 x^{3}+12 c_1 +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_1 \,x^{3}+9 c_1^{2}+12 c_1}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 5.218 (sec). Leaf size: 409
ode=(x^3+2*y[x]-y[x]^2)*D[y[x],x]+3*x^2*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt [3]{2} \left (x^3+1\right )}{\sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}-\frac {\sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{\sqrt [3]{2}}+1\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3+1\right )}{2^{2/3} \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{2 \sqrt [3]{2}}+1\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3+1\right )}{2^{2/3} \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{2 \sqrt [3]{2}}+1\\ y(x)&\to 0 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2*y(x) + (x**3 - y(x)**2 + 2*y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out