83.22.19 problem 19
Internal
problem
ID
[19194]
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
:
19
Date
solved
:
Thursday, March 13, 2025 at 01:50:12 PM
CAS
classification
:
[_quadrature]
\begin{align*} y&=a y^{\prime }+\sqrt {1+{y^{\prime }}^{2}} \end{align*}
✓ Maple. Time used: 0.044 (sec). Leaf size: 112
ode:=y(x) = a*diff(y(x),x)+(1+diff(y(x),x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
\left (\int _{}^{y \left (x \right )}\frac {1}{-\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}-\int _{}^{y \left (x \right )}\frac {1}{-\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_{1} +x &= 0 \\
-\left (\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}+\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_{1} +x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.972 (sec). Leaf size: 210
ode=y[x]==a*D[y[x],x]+Sqrt[1+D[y[x],x]^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {a \left (\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a+1\right )+\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}+a-1\right )\right )-(a+1) \log \left ((a-1) \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}\right )\right )}{a^2-1}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {a \left (\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a-1\right )+\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}+a+1\right )\right )-(a-1) \log \left ((a+1) \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}\right )\right )}{a^2-1}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\
y(x)\to 1 \\
\end{align*}
✓ Sympy. Time used: 1.594 (sec). Leaf size: 58
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*Derivative(y(x), x) - sqrt(Derivative(y(x), x)**2 + 1) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{y a - \sqrt {y^{2} + a^{2} - 1}}\, dy = C_{1} + \frac {x}{\left (a - 1\right ) \left (a + 1\right )}, \ \int \limits ^{y{\left (x \right )}} \frac {1}{y a + \sqrt {y^{2} + a^{2} - 1}}\, dy = C_{1} + \frac {x}{\left (a - 1\right ) \left (a + 1\right )}\right ]
\]