29.25.16 problem 713
Internal
problem
ID
[5303]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
25
Problem
number
:
713
Date
solved
:
Tuesday, March 04, 2025 at 09:19:00 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 2 x \left (x^{3}+y^{4}\right ) y^{\prime }&=\left (x^{3}+2 y^{4}\right ) y \end{align*}
✓ Maple. Time used: 0.316 (sec). Leaf size: 281
ode:=2*x*(x^3+y(x)^4)*diff(y(x),x) = (x^3+2*y(x)^4)*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= -\frac {2^{{3}/{4}} \left (\left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\
y \left (x \right ) &= \frac {2^{{3}/{4}} \left (\left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\
y \left (x \right ) &= -\frac {2^{{3}/{4}} \left (\left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\
y \left (x \right ) &= \frac {2^{{3}/{4}} \left (\left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\
y \left (x \right ) &= -\frac {i 2^{{3}/{4}} \left (\left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\
y \left (x \right ) &= -\frac {i 2^{{3}/{4}} \left (\left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\
y \left (x \right ) &= \frac {i 2^{{3}/{4}} \left (\left (2 c_{1} +x -\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\
y \left (x \right ) &= \frac {i 2^{{3}/{4}} \left (\left (2 c_{1} +x +\sqrt {x \left (4 c_{1} +x \right )}\right ) x^{3} c_{1}^{3}\right )^{{1}/{4}}}{2 c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 3.898 (sec). Leaf size: 166
ode=2 x(x^3+y[x]^4)D[y[x],x]==(x^3+2 y[x]^4)y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {c_1 x^2-x^{3/2} \sqrt {4+c_1{}^2 x}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {c_1 x^2-x^{3/2} \sqrt {4+c_1{}^2 x}}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {x^{3/2} \sqrt {4+c_1{}^2 x}+c_1 x^2}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {x^{3/2} \sqrt {4+c_1{}^2 x}+c_1 x^2}}{\sqrt {2}} \\
y(x)\to 0 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*(x**3 + y(x)**4)*Derivative(y(x), x) - (x**3 + 2*y(x)**4)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (x**3 + 2*y(x)**4)*y(x)/(2*x*(x**3 + y(x)**4)) cannot be solved by the factorable group method