82.11.1 problem Ex. 1
Internal
problem
ID
[18761]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
II.
Equations
of
the
first
order
and
of
the
first
degree.
Exercises
at
page
28
Problem
number
:
Ex.
1
Date
solved
:
Tuesday, January 28, 2025 at 12:15:22 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, _Bernoulli]
\begin{align*} y^{\prime }+\frac {y}{x}&=x^{2} y^{6} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 254
dsolve(diff(y(x),x)+1/x*y(x)=x^2*y(x)^6,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {2^{{1}/{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{{1}/{5}}}{2 c_{1} x^{3}+5 x} \\
y \left (x \right ) &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) 2^{{1}/{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{{1}/{5}}}{8 c_{1} x^{3}+20 x} \\
y \left (x \right ) &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right ) 2^{{1}/{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{{1}/{5}}}{8 c_{1} x^{3}+20 x} \\
y \left (x \right ) &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right ) 2^{{1}/{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{{1}/{5}}}{8 c_{1} x^{3}+20 x} \\
y \left (x \right ) &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) 2^{{1}/{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{{1}/{5}}}{8 c_{1} x^{3}+20 x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.041 (sec). Leaf size: 141
DSolve[D[y[x],x]+1/x*y[x]==x^2*y[x]^6,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt [5]{-2}}{\sqrt [5]{x^3 \left (5+2 c_1 x^2\right )}} \\
y(x)\to \frac {1}{\sqrt [5]{\frac {5 x^3}{2}+c_1 x^5}} \\
y(x)\to \frac {(-1)^{2/5}}{\sqrt [5]{\frac {5 x^3}{2}+c_1 x^5}} \\
y(x)\to -\frac {(-1)^{3/5}}{\sqrt [5]{\frac {5 x^3}{2}+c_1 x^5}} \\
y(x)\to \frac {(-1)^{4/5}}{\sqrt [5]{\frac {5 x^3}{2}+c_1 x^5}} \\
y(x)\to 0 \\
\end{align*}