60.1.447 problem 459

Internal problem ID [10461]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 459
Date solved : Wednesday, March 05, 2025 at 10:54:17 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} {\mathrm e}^{-2 x} {y^{\prime }}^{2}-\left (y^{\prime }-1\right )^{2}+{\mathrm e}^{-2 y}&=0 \end{align*}

✓ Maple. Time used: 0.188 (sec). Leaf size: 128

ode:=exp(-2*x)*diff(y(x),x)^2-(diff(y(x),x)-1)^2+exp(-2*y(x)) = 0; 
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 x +2 c_{1}}}\, {\mathrm e}^{2 x}-{\mathrm e}^{2 c_{1}}}{-{\mathrm e}^{2 c_{1} +2 x}+{\mathrm e}^{2 x}+{\mathrm e}^{2 c_{1}}}\right ) \\ y &= 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 x}+{\mathrm e}^{2 c_{1}}}\right ) \\ \end{align*}

✓ Mathematica. Time used: 21.066 (sec). Leaf size: 809

ode=E^(-2*y[x]) - (-1 + D[y[x],x])^2 + D[y[x],x]^2/E^(2*x)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(Derivative(y(x), x) - 1)**2 + exp(-2*y(x)) + exp(-2*x)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out