83.22.6 problem 6
Internal
problem
ID
[19189]
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
:
Monday, March 31, 2025 at 06:52:23 PM
CAS
classification
:
[_quadrature]
\begin{align*} y&=\frac {2 a {y^{\prime }}^{2}}{\left ({y^{\prime }}^{2}+1\right )^{2}} \end{align*}
✓ Maple. Time used: 0.070 (sec). Leaf size: 141
ode:=y(x) = 2*a*diff(y(x),x)^2/(1+diff(y(x),x)^2)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
x -\int _{}^{y}\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}\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}\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}\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*}
✓ Mathematica. Time used: 102.351 (sec). Leaf size: 572
ode=y[x]==(2*a*D[y[x],x]^2)/(D[y[x],x]^2+1)^2;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 12.524 (sec). Leaf size: 109
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-2*a*Derivative(y(x), x)**2/(Derivative(y(x), x)**2 + 1)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {-1 + \frac {a}{y} - \frac {\sqrt {a \left (- 2 y + a\right )}}{y}}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {-1 + \frac {a}{y} - \frac {\sqrt {a \left (- 2 y + a\right )}}{y}}}\, dy = C_{1} + x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {-1 + \frac {a}{y} + \frac {\sqrt {a \left (- 2 y + a\right )}}{y}}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {-1 + \frac {a}{y} + \frac {\sqrt {a \left (- 2 y + a\right )}}{y}}}\, dy = C_{1} + x\right ]
\]