29.24.3 problem 665
Internal
problem
ID
[5256]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
665
Date
solved
:
Sunday, March 30, 2025 at 07:31:57 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x \left (1+x y^{2}\right ) y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.065 (sec). Leaf size: 137
ode:=x*(1+x*y(x)^2)*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {2}\, \sqrt {x c_1 \left (2 c_1 +x -\sqrt {x \left (4 c_1 +x \right )}\right )}}{2 c_1 x} \\
y &= \frac {\sqrt {2}\, \sqrt {x c_1 \left (2 c_1 +x -\sqrt {x \left (4 c_1 +x \right )}\right )}}{2 c_1 x} \\
y &= -\frac {\sqrt {2}\, \sqrt {x c_1 \left (2 c_1 +x +\sqrt {x \left (4 c_1 +x \right )}\right )}}{2 c_1 x} \\
y &= \frac {\sqrt {2}\, \sqrt {x c_1 \left (2 c_1 +x +\sqrt {x \left (4 c_1 +x \right )}\right )}}{2 c_1 x} \\
\end{align*}
✓ Mathematica. Time used: 0.276 (sec). Leaf size: 65
ode=x(1+x y[x]^2)D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{2} \left (c_1-\frac {\sqrt {4+c_1{}^2 x}}{\sqrt {x}}\right ) \\
y(x)\to \frac {1}{2} \left (\frac {\sqrt {4+c_1{}^2 x}}{\sqrt {x}}+c_1\right ) \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 8.327 (sec). Leaf size: 139
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x*y(x)**2 + 1)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} + \frac {2}{x} - \frac {\sqrt {C_{1} x^{3} \left (C_{1} x + 4\right )}}{x^{2}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} + \frac {2}{x} - \frac {\sqrt {C_{1} x^{3} \left (C_{1} x + 4\right )}}{x^{2}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} + \frac {2}{x} + \frac {\sqrt {C_{1} x^{3} \left (C_{1} x + 4\right )}}{x^{2}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} + \frac {2}{x} + \frac {\sqrt {C_{1} x^{3} \left (C_{1} x + 4\right )}}{x^{2}}}}{2}\right ]
\]