83.23.25 problem 25
Internal
problem
ID
[19306]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
V.
Singular
solutions.
Exercise
V
at
page
76
Problem
number
:
25
Date
solved
:
Tuesday, January 28, 2025 at 01:28:45 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
\begin{align*} \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+b^{2}-y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.098 (sec). Leaf size: 82
dsolve((a^2-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+b^2-y(x)^2=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\sqrt {a^{2}-x^{2}}\, b}{a} \\
y \left (x \right ) &= -\frac {\sqrt {a^{2}-x^{2}}\, b}{a} \\
y \left (x \right ) &= c_{1} x -\sqrt {a^{2} c_{1}^{2}+b^{2}} \\
y \left (x \right ) &= c_{1} x +\sqrt {a^{2} c_{1}^{2}+b^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.816 (sec). Leaf size: 419
DSolve[(a^2-x^2)*D[y[x],x]^2+2*x*y[x]*D[y[x],x]+b^2-y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [-\frac {\sqrt {a^2 \left (y(x)^2-b^2\right )} \arctan \left (\frac {\sqrt {y(x)^2-b^2}}{b}\right )}{b \sqrt {y(x)^2-b^2}}-\frac {2 \sqrt {y(x)^2-b^2} \sqrt {-a^2 \left (b^2-y(x)^2\right )} \arctan \left (\frac {b x \sqrt {y(x)^2-b^2}}{y(x) \left (\sqrt {a^2 \left (y(x)^2-b^2\right )}-\sqrt {a^2 \left (y(x)^2-b^2\right )+b^2 x^2}\right )+b^2 x}\right )}{b^3-b y(x)^2}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {\sqrt {a^2 \left (y(x)^2-b^2\right )} \arctan \left (\frac {\sqrt {y(x)^2-b^2}}{b}\right )}{b \sqrt {y(x)^2-b^2}}+\frac {2 \sqrt {y(x)^2-b^2} \sqrt {-a^2 \left (b^2-y(x)^2\right )} \arctan \left (\frac {b x \sqrt {y(x)^2-b^2}}{y(x) \left (\sqrt {a^2 \left (y(x)^2-b^2\right )+b^2 x^2}-\sqrt {a^2 \left (y(x)^2-b^2\right )}\right )+b^2 x}\right )}{b^3-b y(x)^2}&=c_1,y(x)\right ] \\
y(x)\to -\frac {b \sqrt {a^2-x^2}}{a} \\
y(x)\to \frac {b \sqrt {a^2-x^2}}{a} \\
\end{align*}