83.22.6 problem 6
Internal
problem
ID
[19260]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree.
Exercise
IV
(E)
at
page
63
Problem
number
:
6
Date
solved
:
Tuesday, January 28, 2025 at 01:16:32 PM
CAS
classification
:
[_quadrature]
\begin{align*} y&=\frac {2 a {y^{\prime }}^{2}}{\left ({y^{\prime }}^{2}+1\right )^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.055 (sec). Leaf size: 141
dsolve(y(x)=(2*a*diff(y(x),x)^2)/(diff(y(x),x)^2+1)^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 0 \\
x -\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (a -\textit {\_a} +\sqrt {a \left (-2 \textit {\_a} +a \right )}\right )}}d \textit {\_a} -c_{1} &= 0 \\
x -\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (a -\textit {\_a} -\sqrt {a \left (-2 \textit {\_a} +a \right )}\right )}}d \textit {\_a} -c_{1} &= 0 \\
x +\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (a -\textit {\_a} +\sqrt {a \left (-2 \textit {\_a} +a \right )}\right )}}d \textit {\_a} -c_{1} &= 0 \\
x +\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} \left (a -\textit {\_a} -\sqrt {a \left (-2 \textit {\_a} +a \right )}\right )}}d \textit {\_a} -c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 102.351 (sec). Leaf size: 572
DSolve[y[x]==(2*a*D[y[x],x]^2)/(D[y[x],x]^2+1)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\text {$\#$1} a} \left (-\sqrt {a (a-2 \text {$\#$1})}+2 \text {$\#$1}+a\right )-2 a \sqrt {-a \left (\sqrt {a (a-2 \text {$\#$1})}+\text {$\#$1}-a\right )} \arctan \left (\frac {\sqrt {\text {$\#$1} a}}{\sqrt {-a \left (\sqrt {a (a-2 \text {$\#$1})}+\text {$\#$1}-a\right )}}\right )}{2 \sqrt {\text {$\#$1} a} \sqrt {-\frac {\sqrt {a (a-2 \text {$\#$1})}+\text {$\#$1}-a}{\text {$\#$1}}}}\&\right ][-x+c_1] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\text {$\#$1} a} \left (-\sqrt {a (a-2 \text {$\#$1})}+2 \text {$\#$1}+a\right )-2 a \sqrt {-a \left (\sqrt {a (a-2 \text {$\#$1})}+\text {$\#$1}-a\right )} \arctan \left (\frac {\sqrt {\text {$\#$1} a}}{\sqrt {-a \left (\sqrt {a (a-2 \text {$\#$1})}+\text {$\#$1}-a\right )}}\right )}{2 \sqrt {\text {$\#$1} a} \sqrt {-\frac {\sqrt {a (a-2 \text {$\#$1})}+\text {$\#$1}-a}{\text {$\#$1}}}}\&\right ][x+c_1] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\text {$\#$1} a} \left (\sqrt {a (a-2 \text {$\#$1})}+2 \text {$\#$1}+a\right )-2 a \sqrt {a \left (\sqrt {a (a-2 \text {$\#$1})}-\text {$\#$1}+a\right )} \arctan \left (\frac {\sqrt {\text {$\#$1} a}}{\sqrt {a \left (\sqrt {a (a-2 \text {$\#$1})}-\text {$\#$1}+a\right )}}\right )}{2 \sqrt {\frac {\sqrt {a (a-2 \text {$\#$1})}-\text {$\#$1}+a}{\text {$\#$1}}} \sqrt {\text {$\#$1} a}}\&\right ][-x+c_1] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\text {$\#$1} a} \left (\sqrt {a (a-2 \text {$\#$1})}+2 \text {$\#$1}+a\right )-2 a \sqrt {a \left (\sqrt {a (a-2 \text {$\#$1})}-\text {$\#$1}+a\right )} \arctan \left (\frac {\sqrt {\text {$\#$1} a}}{\sqrt {a \left (\sqrt {a (a-2 \text {$\#$1})}-\text {$\#$1}+a\right )}}\right )}{2 \sqrt {\frac {\sqrt {a (a-2 \text {$\#$1})}-\text {$\#$1}+a}{\text {$\#$1}}} \sqrt {\text {$\#$1} a}}\&\right ][x+c_1] \\
y(x)\to 0 \\
\end{align*}