75.7.1 problem 175
Internal
problem
ID
[16723]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
7,
Total
differential
equations.
The
integrating
factor.
Exercises
page
61
Problem
number
:
175
Date
solved
:
Monday, March 31, 2025 at 03:08:04 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} x \left (2 x^{2}+y^{2}\right )+y \left (x^{2}+2 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 125
ode:=x*(2*x^2+y(x)^2)+y(x)*(x^2+2*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {-2 c_1 \,x^{2}-2 \sqrt {-3 x^{4} c_1^{2}+4}}}{2 \sqrt {c_1}} \\
y &= \frac {\sqrt {-2 c_1 \,x^{2}-2 \sqrt {-3 x^{4} c_1^{2}+4}}}{2 \sqrt {c_1}} \\
y &= -\frac {\sqrt {-2 c_1 \,x^{2}+2 \sqrt {-3 x^{4} c_1^{2}+4}}}{2 \sqrt {c_1}} \\
y &= \frac {\sqrt {-2 c_1 \,x^{2}+2 \sqrt {-3 x^{4} c_1^{2}+4}}}{2 \sqrt {c_1}} \\
\end{align*}
✓ Mathematica. Time used: 25.124 (sec). Leaf size: 303
ode=x*(2*x^2+y[x]^2)+y[x]*(x^2+2*y[x]^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {-x^2-\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {-x^2-\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {-x^2+\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {-x^2+\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {-\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {-\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \\
\end{align*}
✓ Sympy. Time used: 4.482 (sec). Leaf size: 109
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(2*x**2 + y(x)**2) + (x**2 + 2*y(x)**2)*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- x^{2} - \sqrt {C_{1} - 3 x^{4}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- x^{2} - \sqrt {C_{1} - 3 x^{4}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- x^{2} + \sqrt {C_{1} - 3 x^{4}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- x^{2} + \sqrt {C_{1} - 3 x^{4}}}}{2}\right ]
\]