56.1.16 problem 16
Internal
problem
ID
[8728]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
16
Date
solved
:
Sunday, March 30, 2025 at 01:25:11 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y^{\prime }&=\frac {-x y-1}{4 x^{3} y-2 x^{2}} \end{align*}
✓ Maple. Time used: 0.210 (sec). Leaf size: 37
ode:=diff(y(x),x) = (-x*y(x)-1)/(4*x^3*y(x)-2*x^2);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\operatorname {RootOf}\left (c_1 \,\textit {\_Z}^{25}-10 c_1 \,\textit {\_Z}^{20}+25 c_1 \,\textit {\_Z}^{15}-16 x^{5}\right )^{5}-1}{4 x}
\]
✓ Mathematica. Time used: 14.178 (sec). Leaf size: 391
ode=D[y[x],x] == (-x*y[x]-1)/(4*x^3*y[x]-2*x^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {Root}\left [64 \text {$\#$1}^5 c_1{}^5 x^5-80 \text {$\#$1}^4 c_1{}^5 x^4-20 \text {$\#$1}^3 c_1{}^5 x^3+25 \text {$\#$1}^2 c_1{}^5 x^2+10 \text {$\#$1} c_1{}^5 x-x^5+c_1{}^5\&,1\right ] \\
y(x)\to \text {Root}\left [64 \text {$\#$1}^5 c_1{}^5 x^5-80 \text {$\#$1}^4 c_1{}^5 x^4-20 \text {$\#$1}^3 c_1{}^5 x^3+25 \text {$\#$1}^2 c_1{}^5 x^2+10 \text {$\#$1} c_1{}^5 x-x^5+c_1{}^5\&,2\right ] \\
y(x)\to \text {Root}\left [64 \text {$\#$1}^5 c_1{}^5 x^5-80 \text {$\#$1}^4 c_1{}^5 x^4-20 \text {$\#$1}^3 c_1{}^5 x^3+25 \text {$\#$1}^2 c_1{}^5 x^2+10 \text {$\#$1} c_1{}^5 x-x^5+c_1{}^5\&,3\right ] \\
y(x)\to \text {Root}\left [64 \text {$\#$1}^5 c_1{}^5 x^5-80 \text {$\#$1}^4 c_1{}^5 x^4-20 \text {$\#$1}^3 c_1{}^5 x^3+25 \text {$\#$1}^2 c_1{}^5 x^2+10 \text {$\#$1} c_1{}^5 x-x^5+c_1{}^5\&,4\right ] \\
y(x)\to \text {Root}\left [64 \text {$\#$1}^5 c_1{}^5 x^5-80 \text {$\#$1}^4 c_1{}^5 x^4-20 \text {$\#$1}^3 c_1{}^5 x^3+25 \text {$\#$1}^2 c_1{}^5 x^2+10 \text {$\#$1} c_1{}^5 x-x^5+c_1{}^5\&,5\right ] \\
\end{align*}
✓ Sympy. Time used: 0.577 (sec). Leaf size: 29
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x*y(x) + 1)/(4*x**3*y(x) - 2*x**2) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
- \log {\left (x \right )} + \frac {2 \log {\left (x y{\left (x \right )} - 1 \right )}}{5} + \frac {3 \log {\left (x y{\left (x \right )} + \frac {1}{4} \right )}}{5} = C_{1}
\]