83.4.19 problem 19
Internal
problem
ID
[19012]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
II.
Equations
of
first
order
and
first
degree.
Exercise
II
(C)
at
page
12
Problem
number
:
19
Date
solved
:
Thursday, March 13, 2025 at 01:21:54 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} x \left (x^{2}+3 y^{2}\right )+y \left (y^{2}+3 x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.056 (sec). Leaf size: 119
ode:=x*(x^2+3*y(x)^2)+y(x)*(y(x)^2+3*x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {\sqrt {-3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\
y \left (x \right ) &= \frac {\sqrt {-3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\
y \left (x \right ) &= -\frac {\sqrt {-3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\
y \left (x \right ) &= -\frac {\sqrt {-3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\
\end{align*}
✓ Mathematica. Time used: 8.389 (sec). Leaf size: 245
ode=x*(x^2+3*y[x]^2)+y[x]*(y[x]^2+3*x^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sqrt {-3 x^2-\sqrt {8 x^4+e^{4 c_1}}} \\
y(x)\to \sqrt {-3 x^2-\sqrt {8 x^4+e^{4 c_1}}} \\
y(x)\to -\sqrt {-3 x^2+\sqrt {8 x^4+e^{4 c_1}}} \\
y(x)\to \sqrt {-3 x^2+\sqrt {8 x^4+e^{4 c_1}}} \\
y(x)\to -\sqrt {-2 \sqrt {2} \sqrt {x^4}-3 x^2} \\
y(x)\to \sqrt {-2 \sqrt {2} \sqrt {x^4}-3 x^2} \\
y(x)\to -\sqrt {2 \sqrt {2} \sqrt {x^4}-3 x^2} \\
y(x)\to \sqrt {2 \sqrt {2} \sqrt {x^4}-3 x^2} \\
\end{align*}
✓ Sympy. Time used: 3.951 (sec). Leaf size: 88
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x**2 + 3*y(x)**2) + (3*x**2 + y(x)**2)*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {- 3 x^{2} - \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = \sqrt {- 3 x^{2} - \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = - \sqrt {- 3 x^{2} + \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = \sqrt {- 3 x^{2} + \sqrt {C_{1} + 8 x^{4}}}\right ]
\]