15.4.16 problem 17
Internal
problem
ID
[2929]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
8,
page
34
Problem
number
:
17
Date
solved
:
Sunday, March 30, 2025 at 12:57:31 AM
CAS
classification
:
[[_homogeneous, `class G`], _exact, _rational]
\begin{align*} \frac {2 y}{x^{3}}+\frac {2 x}{y^{2}}&=\left (\frac {1}{x^{2}}+\frac {2 x^{2}}{y^{3}}\right ) y^{\prime } \end{align*}
✓ Maple. Time used: 0.365 (sec). Leaf size: 231
ode:=2*y(x)/x^3+2*x/y(x)^2 = (1/x^2+2*x^2/y(x)^3)*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {x^{{4}/{3}} \operatorname {RootOf}\left (x^{14} \textit {\_Z}^{18}+6 x^{{38}/{3}} \textit {\_Z}^{16}+15 x^{{34}/{3}} \textit {\_Z}^{14}+\left (20 x^{10}-27 c_1 \,x^{8}\right ) \textit {\_Z}^{12}+\left (15 x^{{26}/{3}}-81 x^{{20}/{3}} c_1 \right ) \textit {\_Z}^{10}+\left (6 x^{{22}/{3}}-108 x^{{16}/{3}} c_1 \right ) \textit {\_Z}^{8}+\left (x^{6}-81 c_1 \,x^{4}\right ) \textit {\_Z}^{6}-36 x^{{8}/{3}} c_1 \,\textit {\_Z}^{4}-9 x^{{4}/{3}} c_1 \,\textit {\_Z}^{2}-c_1 \right )^{2}+1}{\operatorname {RootOf}\left (x^{14} \textit {\_Z}^{18}+6 x^{{38}/{3}} \textit {\_Z}^{16}+15 x^{{34}/{3}} \textit {\_Z}^{14}+\left (20 x^{10}-27 c_1 \,x^{8}\right ) \textit {\_Z}^{12}+\left (15 x^{{26}/{3}}-81 x^{{20}/{3}} c_1 \right ) \textit {\_Z}^{10}+\left (6 x^{{22}/{3}}-108 x^{{16}/{3}} c_1 \right ) \textit {\_Z}^{8}+\left (x^{6}-81 c_1 \,x^{4}\right ) \textit {\_Z}^{6}-36 x^{{8}/{3}} c_1 \,\textit {\_Z}^{4}-9 x^{{4}/{3}} c_1 \,\textit {\_Z}^{2}-c_1 \right )^{2}}
\]
✓ Mathematica. Time used: 12.233 (sec). Leaf size: 414
ode=2*(y[x]/x^3+x/y[x]^2)==(1/x^2+2*x^2/y[x]^3)*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{3} \left (c_1 x^2+\frac {c_1{}^2 x^4}{\sqrt [3]{c_1{}^3 x^6+\frac {27 x^4}{2}+\frac {3}{2} \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}}+\sqrt [3]{c_1{}^3 x^6+\frac {27 x^4}{2}+\frac {3}{2} \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}\right ) \\
y(x)\to \frac {1}{12} \left (4 c_1 x^2-\frac {2 \left (1+i \sqrt {3}\right ) c_1{}^2 x^4}{\sqrt [3]{c_1{}^3 x^6+\frac {27 x^4}{2}+\frac {3}{2} \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{2 c_1{}^3 x^6+27 x^4+3 \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}\right ) \\
y(x)\to \frac {1}{12} \left (4 c_1 x^2+\frac {2 i \left (\sqrt {3}+i\right ) c_1{}^2 x^4}{\sqrt [3]{c_1{}^3 x^6+\frac {27 x^4}{2}+\frac {3}{2} \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{2 c_1{}^3 x^6+27 x^4+3 \sqrt {3} \sqrt {x^8 \left (27+4 c_1{}^3 x^2\right )}}\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x/y(x)**2 - (2*x**2/y(x)**3 + x**(-2))*Derivative(y(x), x) + 2*y(x)/x**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - 2*(x**4 + y(x)**3)*y(x)/(x*(2*x**4 + y(x)**3)) cannot be solved by the factorable group method