83.23.22 problem 22
Internal
problem
ID
[19303]
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
:
22
Date
solved
:
Tuesday, January 28, 2025 at 01:25:28 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} \left (1-y^{\prime }\right )^{2}-{\mathrm e}^{-2 y}&={y^{\prime }}^{2} {\mathrm e}^{-2 x} \end{align*}
✓ Solution by Maple
Time used: 0.105 (sec). Leaf size: 128
dsolve((1-diff(y(x),x))^2-exp(-2*y(x))=diff(y(x),x)^2*exp(-2*x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= c_{1} -\ln \left (\frac {\sqrt {{\mathrm e}^{-2 x +4 c_{1}}-{\mathrm e}^{-2 x +2 c_{1}}}\, {\mathrm e}^{2 x}-{\mathrm e}^{2 c_{1}}}{-{\mathrm e}^{2 c_{1} +2 x}+{\mathrm e}^{2 c_{1}}+{\mathrm e}^{2 x}}\right ) \\
y \left (x \right ) &= c_{1} -\ln \left (\frac {-\sqrt {{\mathrm e}^{-2 x +4 c_{1}}-{\mathrm e}^{-2 x +2 c_{1}}}\, {\mathrm e}^{2 x}-{\mathrm e}^{2 c_{1}}}{-{\mathrm e}^{2 c_{1} +2 x}+{\mathrm e}^{2 c_{1}}+{\mathrm e}^{2 x}}\right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 70.160 (sec). Leaf size: 377
DSolve[(1-D[y[x],x])^2-Exp[-2*y[x]]==D[y[x],x]^2*Exp[-2*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \log \left (-\frac {i \left (e^x-1\right ) e^{c_1 \left (-e^{-x}\right ) \sqrt {e^{2 x}}} \left (e^x+e^{2 c_1 e^{-x} \sqrt {e^{2 x}}}+e^{x+2 c_1 e^{-x} \sqrt {e^{2 x}}}-1\right )}{2 \sqrt {\left (e^x-1\right )^2}}\right ) \\
y(x)\to \log \left (\frac {i \left (e^x-1\right ) e^{c_1 \left (-e^{-x}\right ) \sqrt {e^{2 x}}} \left (e^x+e^{2 c_1 e^{-x} \sqrt {e^{2 x}}}+e^{x+2 c_1 e^{-x} \sqrt {e^{2 x}}}-1\right )}{2 \sqrt {\left (e^x-1\right )^2}}\right ) \\
y(x)\to \log \left (-\frac {i \left (e^x+1\right ) e^{c_1 \left (-e^{-x}\right ) \sqrt {e^{2 x}}} \left (e^x+e^{2 c_1 e^{-x} \sqrt {e^{2 x}}}+e^{x+2 c_1 e^{-x} \sqrt {e^{2 x}}}-1\right )}{2 \sqrt {\left (e^x+1\right )^2}}\right ) \\
y(x)\to \log \left (\frac {i \left (e^x+1\right ) e^{c_1 \left (-e^{-x}\right ) \sqrt {e^{2 x}}} \left (e^x+e^{2 c_1 e^{-x} \sqrt {e^{2 x}}}+e^{x+2 c_1 e^{-x} \sqrt {e^{2 x}}}-1\right )}{2 \sqrt {\left (e^x+1\right )^2}}\right ) \\
\end{align*}