82.11.1 problem Ex. 1
Internal
problem
ID
[18690]
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
:
Monday, March 31, 2025 at 05:57:43 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, _Bernoulli]
\begin{align*} y^{\prime }+\frac {y}{x}&=x^{2} y^{6} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 254
ode:=diff(y(x),x)+y(x)/x = x^2*y(x)^6;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \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 &= -\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 &= \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 &= -\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 &= \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*}
✓ Mathematica. Time used: 1.041 (sec). Leaf size: 141
ode=D[y[x],x]+1/x*y[x]==x^2*y[x]^6;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 12.722 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2*y(x)**6 + Derivative(y(x), x) + y(x)/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [5]{2} \sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}}, \ y{\left (x \right )} = \frac {\sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}} \left (- \sqrt [5]{2} + \sqrt [5]{2} \sqrt {5} - 2^{\frac {7}{10}} i \sqrt {\sqrt {5} + 5}\right )}{4}, \ y{\left (x \right )} = \frac {\sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}} \left (- \sqrt [5]{2} + \sqrt [5]{2} \sqrt {5} + 2^{\frac {7}{10}} i \sqrt {\sqrt {5} + 5}\right )}{4}, \ y{\left (x \right )} = \frac {\sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}} \left (- \sqrt [5]{2} \sqrt {5} - \sqrt [5]{2} - 2^{\frac {7}{10}} i \sqrt {5 - \sqrt {5}}\right )}{4}, \ y{\left (x \right )} = \frac {\sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}} \left (- \sqrt [5]{2} \sqrt {5} - \sqrt [5]{2} + 2^{\frac {7}{10}} i \sqrt {5 - \sqrt {5}}\right )}{4}\right ]
\]