40.5.8 problem 24
Internal
problem
ID
[6673]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
9.
Equations
of
first
order
and
higher
degree.
Supplemetary
problems.
Page
65
Problem
number
:
24
Date
solved
:
Monday, January 27, 2025 at 02:18:54 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational]
\begin{align*} 16 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.106 (sec). Leaf size: 105
dsolve(16*y(x)^3*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*}
y &= -\frac {\sqrt {2}\, \sqrt {-x}}{2} \\
y &= \frac {\sqrt {2}\, \sqrt {-x}}{2} \\
y &= -\frac {\sqrt {x}\, \sqrt {2}}{2} \\
y &= \frac {\sqrt {x}\, \sqrt {2}}{2} \\
y &= 0 \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a}^{4}-\sqrt {-4 \textit {\_a}^{4}+1}-1}{\textit {\_a} \left (4 \textit {\_a}^{4}-1\right )}d \textit {\_a} \right )+c_1 \right ) \sqrt {x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.539 (sec). Leaf size: 307
DSolve[16*y[x]^3*D[y[x],x]^2-4*x*D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt [4]{2} \sqrt [4]{e^{c_1} \left (x-2 e^{c_1}\right )} \\
y(x)\to -i \sqrt [4]{2} \sqrt [4]{e^{c_1} \left (x-2 e^{c_1}\right )} \\
y(x)\to i \sqrt [4]{2} \sqrt [4]{e^{c_1} \left (x-2 e^{c_1}\right )} \\
y(x)\to \sqrt [4]{2} \sqrt [4]{e^{c_1} \left (x-2 e^{c_1}\right )} \\
y(x)\to -\frac {\sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{\sqrt {2}} \\
y(x)\to -\frac {i \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{\sqrt {2}} \\
y(x)\to \frac {i \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{\sqrt {2}} \\
y(x)\to 0 \\
y(x)\to -\frac {\sqrt {x}}{\sqrt {2}} \\
y(x)\to -\frac {i \sqrt {x}}{\sqrt {2}} \\
y(x)\to \frac {i \sqrt {x}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {x}}{\sqrt {2}} \\
\end{align*}