23.1.610 problem 604
Internal
problem
ID
[5217]
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
:
604
Date
solved
:
Tuesday, September 30, 2025 at 11:56:06 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} \left (x^{2}-y^{2}\right ) y^{\prime }+x \left (x +2 y\right )&=0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 320
ode:=(x^2-y(x)^2)*diff(y(x),x)+x*(x+2*y(x)) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {4 c_1 \,x^{2}+\left (4+4 x^{3} c_1^{{3}/{2}}+4 \sqrt {-3 x^{6} c_1^{3}+2 x^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}}}{2 \left (4+4 x^{3} c_1^{{3}/{2}}+4 \sqrt {-3 x^{6} c_1^{3}+2 x^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= \frac {4 i \sqrt {3}\, c_1 \,x^{2}-i \sqrt {3}\, \left (4+4 x^{3} c_1^{{3}/{2}}+4 \sqrt {-3 x^{6} c_1^{3}+2 x^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}}-4 c_1 \,x^{2}-\left (4+4 x^{3} c_1^{{3}/{2}}+4 \sqrt {-3 x^{6} c_1^{3}+2 x^{3} c_1^{{3}/{2}}+1}\right )^{{2}/{3}}}{4 \left (4+4 x^{3} c_1^{{3}/{2}}+4 \sqrt {-3 x^{6} c_1^{3}+2 x^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_1}} \\
y &= \frac {\left (4+4 x^{3} c_1^{{3}/{2}}+4 \sqrt {-3 x^{6} c_1^{3}+2 x^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 \sqrt {c_1}}-\frac {\left (1+i \sqrt {3}\right ) x^{2} \sqrt {c_1}}{\left (4+4 x^{3} c_1^{{3}/{2}}+4 \sqrt {-3 x^{6} c_1^{3}+2 x^{3} c_1^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 60.131 (sec). Leaf size: 359
ode=(x^2-y[x]^2)*D[y[x],x]+x*(x+2*y[x])==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}\\ y(x)&\to \frac {i \left (\sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}-2 \left (\sqrt {3}-i\right ) x^2\right )}{2\ 2^{2/3} \sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}\\ y(x)&\to \frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2+2^{2/3} \left (-1-i \sqrt {3}\right ) \left (x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{x^3+\sqrt {-3 x^6+2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x + 2*y(x)) + (x**2 - y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out