54.2.12 problem 14
Internal
problem
ID
[8544]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
16.
Nonlinear
equations.
Miscellaneous
Exercises.
Page
340
Problem
number
:
14
Date
solved
:
Sunday, March 30, 2025 at 01:17:18 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} 16 x {y^{\prime }}^{2}+8 y y^{\prime }+y^{6}&=0 \end{align*}
✓ Maple. Time used: 0.053 (sec). Leaf size: 101
ode:=16*x*diff(y(x),x)^2+8*y(x)*diff(y(x),x)+y(x)^6 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {1}{x^{{1}/{4}}} \\
y &= -\frac {1}{x^{{1}/{4}}} \\
y &= -\frac {i}{x^{{1}/{4}}} \\
y &= \frac {i}{x^{{1}/{4}}} \\
y &= 0 \\
y &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_1 +4 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {-\textit {\_a}^{4}+1}}d \textit {\_a} \right )}{x^{{1}/{4}}} \\
y &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_1 -4 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {-\textit {\_a}^{4}+1}}d \textit {\_a} \right )}{x^{{1}/{4}}} \\
\end{align*}
✓ Mathematica. Time used: 0.608 (sec). Leaf size: 171
ode=16*x*(D[y[x],x])^2+8*y[x]*D[y[x],x]+y[x]^6==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}} \\
y(x)\to -\frac {i \sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}} \\
y(x)\to \frac {i \sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}} \\
y(x)\to \frac {\sqrt {2} e^{\frac {c_1}{4}}}{\sqrt {x+e^{c_1}}} \\
y(x)\to 0 \\
y(x)\to -\frac {1}{\sqrt [4]{x}} \\
y(x)\to -\frac {i}{\sqrt [4]{x}} \\
y(x)\to \frac {i}{\sqrt [4]{x}} \\
y(x)\to \frac {1}{\sqrt [4]{x}} \\
\end{align*}
✓ Sympy. Time used: 1.308 (sec). Leaf size: 92
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(16*x*Derivative(y(x), x)**2 + y(x)**6 + 8*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} 2 \operatorname {acosh}{\left (\frac {1}{\sqrt {x} y^{2}{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{x y^{4}{\left (x \right )}}\right |} > 1 \\- 2 i \operatorname {asin}{\left (\frac {1}{\sqrt {x} y^{2}{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = C_{1} + \log {\left (x \right )}, \ \begin {cases} - 2 \operatorname {acosh}{\left (\frac {1}{\sqrt {x} y^{2}{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{x y^{4}{\left (x \right )}}\right |} > 1 \\2 i \operatorname {asin}{\left (\frac {1}{\sqrt {x} y^{2}{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = C_{1} + \log {\left (x \right )}\right ]
\]