68.8.6 problem 6
Internal
problem
ID
[17429]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Review
exercises,
page
80
Problem
number
:
6
Date
solved
:
Thursday, October 02, 2025 at 02:20:45 PM
CAS
classification
:
[_separable]
\begin{align*} -\frac {1}{x^{5}}+\frac {1}{x^{3}}&=\left (2 y^{4}-6 y^{9}\right ) y^{\prime } \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 522
ode:=-1/x^5+1/x^3 = (2*y(x)^4-6*y(x)^9)*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {6^{{4}/{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{6 x} \\
y &= \frac {6^{{4}/{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{6 x} \\
y &= -\frac {\left (i \sqrt {10-2 \sqrt {5}}+\sqrt {5}+1\right ) 6^{{4}/{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{24 x} \\
y &= \frac {\left (i \sqrt {10-2 \sqrt {5}}-\sqrt {5}-1\right ) 6^{{4}/{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{24 x} \\
y &= -\frac {\left (i \sqrt {10+2 \sqrt {5}}-\sqrt {5}+1\right ) 6^{{4}/{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{24 x} \\
y &= \frac {\left (i \sqrt {10+2 \sqrt {5}}+\sqrt {5}-1\right ) 6^{{4}/{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{24 x} \\
y &= -\frac {\left (i \sqrt {10-2 \sqrt {5}}+\sqrt {5}+1\right ) 6^{{4}/{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{24 x} \\
y &= \frac {\left (i \sqrt {10-2 \sqrt {5}}-\sqrt {5}-1\right ) 6^{{4}/{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{24 x} \\
y &= -\frac {\left (i \sqrt {10+2 \sqrt {5}}-\sqrt {5}+1\right ) 6^{{4}/{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{24 x} \\
y &= \frac {\left (i \sqrt {10+2 \sqrt {5}}+\sqrt {5}-1\right ) 6^{{4}/{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_1 \,x^{4}+30 x^{2}-15}\right )^{{1}/{5}}}{24 x} \\
\end{align*}
✓ Mathematica. Time used: 0.214 (sec). Leaf size: 45
ode=(-x^(-5)+x^(-3))==(2*y[x]^4-6*y[x]^9)*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}K[1]^4 \left (3 K[1]^5-1\right )dK[1]\&\right ]\left [\frac {2 x^2-1}{8 x^4}+c_1\right ] \end{align*}
✓ Sympy. Time used: 62.014 (sec). Leaf size: 649
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((6*y(x)**9 - 2*y(x)**4)*Derivative(y(x), x) + x**(-3) - 1/x**5,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\text {Solution too large to show}
\]