40.4.8 problem 19 (i)
Internal
problem
ID
[6648]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
6.
Equations
of
first
order
and
first
degree
(Linear
equations).
Supplemetary
problems.
Page
39
Problem
number
:
19
(i)
Date
solved
:
Sunday, March 30, 2025 at 11:13:31 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational, _Bernoulli]
\begin{align*} x y^{\prime }+y-x^{3} y^{6}&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 254
ode:=x*diff(y(x),x)+y(x)-x^3*y(x)^6 = 0;
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: 1.089 (sec). Leaf size: 141
ode=x*D[y[x],x]+(y[x]-x^3*y[x]^6)==0;
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: 15.295 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**3*y(x)**6 + x*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [5]{2} \sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}}, \ y{\left (x \right )} = \frac {\sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}} \left (- \sqrt [5]{2} + \sqrt [5]{2} \sqrt {5} - 2^{\frac {7}{10}} i \sqrt {\sqrt {5} + 5}\right )}{4}, \ y{\left (x \right )} = \frac {\sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}} \left (- \sqrt [5]{2} + \sqrt [5]{2} \sqrt {5} + 2^{\frac {7}{10}} i \sqrt {\sqrt {5} + 5}\right )}{4}, \ y{\left (x \right )} = \frac {\sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}} \left (- \sqrt [5]{2} \sqrt {5} - \sqrt [5]{2} - 2^{\frac {7}{10}} i \sqrt {5 - \sqrt {5}}\right )}{4}, \ y{\left (x \right )} = \frac {\sqrt [5]{\frac {1}{x^{3} \left (C_{1} x^{2} + 5\right )}} \left (- \sqrt [5]{2} \sqrt {5} - \sqrt [5]{2} + 2^{\frac {7}{10}} i \sqrt {5 - \sqrt {5}}\right )}{4}\right ]
\]