74.8.6 problem 6
Internal
problem
ID
[16142]
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
:
Tuesday, January 28, 2025 at 08:51:52 AM
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*}
✓ Solution by Maple
Time used: 0.053 (sec). Leaf size: 514
dsolve((-x^(-5)+x^(-3))=(2*y(x)^4-6*y(x)^9)*diff(y(x),x),y(x), singsol=all)
\begin{align*}
y &= \frac {6^{{4}/{5}} \left (2 x^{5}+\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\right )^{{1}/{5}}}{6 x} \\
y &= \frac {6^{{4}/{5}} \left (2 x^{5}-\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\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}+\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\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}+\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\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}+\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\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}+\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\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}-\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\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}-\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\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}-\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\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}-\sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\, x^{3}\right )^{{1}/{5}}}{24 x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.347 (sec). Leaf size: 45
DSolve[(-x^(-5)+x^(-3))==(2*y[x]^4-6*y[x]^9)*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}K[1]^4 \left (3 K[1]^5-1\right )dK[1]\&\right ]\left [\frac {2 x^2-1}{8 x^4}+c_1\right ]
\]