44.6.15 problem 15
Internal
problem
ID
[7159]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.3
Linear
equations.
Exercises
2.3
at
page
63
Problem
number
:
15
Date
solved
:
Tuesday, February 04, 2025 at 12:41:34 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y-4 \left (x +y^{6}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 989
dsolve(y(x)-4*(x+y(x)^6)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= -\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}} \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{2}/{3}}-c_1 \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}+c_1^{2}\right )}}{6 \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}} \\
y &= \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}} \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{2}/{3}}-c_1 \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}+c_1^{2}\right )}}{6 \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}} \\
y &= -\frac {\sqrt {3}\, \sqrt {-\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}} \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}+c_1 \right ) \left (i \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}-c_1 \right ) \sqrt {3}+\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}+c_1 \right )}}{6 \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}} \\
y &= \frac {\sqrt {3}\, \sqrt {-\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}} \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}+c_1 \right ) \left (i \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}-c_1 \right ) \sqrt {3}+\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}+c_1 \right )}}{6 \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}} \\
y &= -\frac {\sqrt {3}\, \sqrt {\left (i \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}-c_1 \right ) \sqrt {3}-\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}-c_1 \right ) \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}} \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}+c_1 \right )}}{6 \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}} \\
y &= \frac {\sqrt {3}\, \sqrt {\left (i \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}-c_1 \right ) \sqrt {3}-\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}-c_1 \right ) \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}} \left (\left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}+c_1 \right )}}{6 \left (-c_1^{3}+6 \sqrt {3}\, \sqrt {-x \,c_1^{3}+27 x^{2}}+54 x \right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 35.293 (sec). Leaf size: 827
DSolve[y[x]-4*(x+y[x]^6)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {\frac {c_1{}^2}{\sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}}+\sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}-c_1}}{\sqrt {6}} \\
y(x)\to \frac {\sqrt {\frac {c_1{}^2}{\sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}}+\sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}-c_1}}{\sqrt {6}} \\
y(x)\to -\frac {\sqrt {-\frac {i \left (\sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}+c_1\right ) \left (\left (\sqrt {3}-i\right ) \sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}-\left (\sqrt {3}+i\right ) c_1\right )}{\sqrt [3]{54 x+6 \sqrt {3} \sqrt {x \left (27 x-c_1{}^3\right )}-c_1{}^3}}}}{2 \sqrt {3}} \\
y(x)\to \frac {\sqrt {-\frac {i \left (\sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}+c_1\right ) \left (\left (\sqrt {3}-i\right ) \sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}-\left (\sqrt {3}+i\right ) c_1\right )}{\sqrt [3]{54 x+6 \sqrt {3} \sqrt {x \left (27 x-c_1{}^3\right )}-c_1{}^3}}}}{2 \sqrt {3}} \\
y(x)\to -\frac {\sqrt {\frac {i \left (\sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}+c_1\right ) \left (\left (\sqrt {3}+i\right ) \sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}-\left (\sqrt {3}-i\right ) c_1\right )}{\sqrt [3]{54 x+6 \sqrt {3} \sqrt {x \left (27 x-c_1{}^3\right )}-c_1{}^3}}}}{2 \sqrt {3}} \\
y(x)\to \frac {\sqrt {\frac {i \left (\sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}+c_1\right ) \left (\left (\sqrt {3}+i\right ) \sqrt [3]{54 x+6 \sqrt {3} \sqrt {-x \left (-27 x+c_1{}^3\right )}-c_1{}^3}-\left (\sqrt {3}-i\right ) c_1\right )}{\sqrt [3]{54 x+6 \sqrt {3} \sqrt {x \left (27 x-c_1{}^3\right )}-c_1{}^3}}}}{2 \sqrt {3}} \\
y(x)\to 0 \\
\end{align*}