29.23.19 problem 650
Internal
problem
ID
[5241]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
23
Problem
number
:
650
Date
solved
:
Sunday, March 30, 2025 at 07:19:45 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x \left (x^{2}+2 y^{2}\right ) y^{\prime }&=\left (2 x^{2}+3 y^{2}\right ) y \end{align*}
✓ Maple. Time used: 0.248 (sec). Leaf size: 89
ode:=x*(x^2+2*y(x)^2)*diff(y(x),x) = (2*x^2+3*y(x)^2)*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {-2-2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\
y &= \frac {\sqrt {-2-2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\
y &= -\frac {\sqrt {-2+2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\
y &= \frac {\sqrt {-2+2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\
\end{align*}
✓ Mathematica. Time used: 43.871 (sec). Leaf size: 277
ode=x(x^2+2 y[x]^2)D[y[x],x]==(2 x^2+3 y[x]^2)y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {-x^2+\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}} \\
y(x)\to \sqrt {-\frac {x^2}{2}+\frac {1}{2} \sqrt {x^4+4 e^{2 c_1} x^6}} \\
y(x)\to -\frac {\sqrt {-\sqrt {x^4}-x^2}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {-\sqrt {x^4}-x^2}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {\sqrt {x^4}-x^2}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {\sqrt {x^4}-x^2}}{\sqrt {2}} \\
\end{align*}
✓ Sympy. Time used: 7.544 (sec). Leaf size: 116
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x**2 + 2*y(x)**2)*Derivative(y(x), x) - (2*x**2 + 3*y(x)**2)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} \left (\sqrt {C_{1} x^{2} + 1} - 1\right )}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} \left (\sqrt {C_{1} x^{2} + 1} - 1\right )}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{2} \left (- \sqrt {C_{1} x^{2} + 1} - 1\right )}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{2} \left (- \sqrt {C_{1} x^{2} + 1} - 1\right )}}{2}\right ]
\]