83.23.22 problem 22
Internal
problem
ID
[19232]
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
:
Monday, March 31, 2025 at 06:59:44 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*}
✓ Maple. Time used: 0.208 (sec). Leaf size: 128
ode:=(1-diff(y(x),x))^2-exp(-2*y(x)) = diff(y(x),x)^2*exp(-2*x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= c_1 -\ln \left (\frac {\sqrt {{\mathrm e}^{-2 x +4 c_1}-{\mathrm e}^{2 c_1 -2 x}}\, {\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 &= c_1 -\ln \left (\frac {-\sqrt {{\mathrm e}^{-2 x +4 c_1}-{\mathrm e}^{2 c_1 -2 x}}\, {\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*}
✓ Mathematica. Time used: 70.16 (sec). Leaf size: 377
ode=(1-D[y[x],x])^2-Exp[-2*y[x]]==D[y[x],x]^2*Exp[-2*x];
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((1 - Derivative(y(x), x))**2 - exp(-2*y(x)) - exp(-2*x)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out