77.23.22 problem 22

Internal problem ID [20588]
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 : Sunday, October 12, 2025 at 05:48:13 AM
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: 68.202 (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