29.24.12 problem 674
Internal
problem
ID
[5265]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
674
Date
solved
:
Tuesday, March 04, 2025 at 08:58:39 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} \left (1-x^{4} y^{2}\right ) y^{\prime }&=x^{3} y^{3} \end{align*}
✓ Maple. Time used: 0.941 (sec). Leaf size: 157
ode:=(1-x^4*y(x)^2)*diff(y(x),x) = x^3*y(x)^3;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {\sqrt {-c_{1} -\sqrt {c_{1} \left (x^{4}+c_{1} \right )}}\, \left (c_{1} -\sqrt {c_{1} \left (x^{4}+c_{1} \right )}\right )}{c_{1} x^{4}} \\
y \left (x \right ) &= \frac {\sqrt {-c_{1} +\sqrt {c_{1} \left (x^{4}+c_{1} \right )}}\, \left (c_{1} +\sqrt {c_{1} \left (x^{4}+c_{1} \right )}\right )}{c_{1} x^{4}} \\
y \left (x \right ) &= \frac {\sqrt {-c_{1} -\sqrt {c_{1} \left (x^{4}+c_{1} \right )}}\, \left (-c_{1} +\sqrt {c_{1} \left (x^{4}+c_{1} \right )}\right )}{c_{1} x^{4}} \\
y \left (x \right ) &= \frac {\left (-c_{1} -\sqrt {c_{1} \left (x^{4}+c_{1} \right )}\right ) \sqrt {-c_{1} +\sqrt {c_{1} \left (x^{4}+c_{1} \right )}}}{c_{1} x^{4}} \\
\end{align*}
✓ Mathematica. Time used: 13.616 (sec). Leaf size: 122
ode=(1-x^4 y[x]^2)D[y[x],x]==x^3 y[x]^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sqrt {\frac {1-\sqrt {1+4 c_1 x^4}}{x^4}} \\
y(x)\to \sqrt {\frac {1-\sqrt {1+4 c_1 x^4}}{x^4}} \\
y(x)\to -\sqrt {\frac {1+\sqrt {1+4 c_1 x^4}}{x^4}} \\
y(x)\to \sqrt {\frac {1+\sqrt {1+4 c_1 x^4}}{x^4}} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 4.459 (sec). Leaf size: 85
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**3*y(x)**3 + (-x**4*y(x)**2 + 1)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt {\frac {1 - \sqrt {C_{1} x^{4} + 1}}{x^{4}}}, \ y{\left (x \right )} = - \sqrt {\frac {\sqrt {C_{1} x^{4} + 1} + 1}{x^{4}}}, \ y{\left (x \right )} = \sqrt {\frac {\sqrt {C_{1} x^{4} + 1} + 1}{x^{4}}}, \ y{\left (x \right )} = - \sqrt {\frac {1 - \sqrt {C_{1} x^{4} + 1}}{x^{4}}}\right ]
\]