23.2.89 problem 91

Internal problem ID [5444]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 91
Date solved : Tuesday, September 30, 2025 at 12:43:27 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} 4 {y^{\prime }}^{2}+2 x \,{\mathrm e}^{-2 y} y^{\prime }-{\mathrm e}^{-2 y}&=0 \end{align*}

✓ Maple. Time used: 0.078 (sec). Leaf size: 113

ode:=4*diff(y(x),x)^2+2*x*exp(-2*y(x))*diff(y(x),x)-exp(-2*y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\ln \left (2\right )-\frac {\ln \left (-\frac {1}{x^{2}}\right )}{2} \\ y &= c_1 -\operatorname {arctanh}\left (\frac {x}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-x^{2}-4 \operatorname {RootOf}\left (-x^{2} {\mathrm e}^{2 c_1}+\textit {\_Z}^{2}-2 \textit {\_Z} \,{\mathrm e}^{2 c_1}+{\mathrm e}^{4 c_1}\right )\right )}\right ) \\ y &= c_1 +\operatorname {arctanh}\left (\frac {x}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-x^{2}-4 \operatorname {RootOf}\left (-x^{2} {\mathrm e}^{2 c_1}+\textit {\_Z}^{2}-2 \textit {\_Z} \,{\mathrm e}^{2 c_1}+{\mathrm e}^{4 c_1}\right )\right )}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 77

ode=4 (D[y[x],x])^2+2 x Exp[-2 y[x]] D[y[x],x]-Exp[-2 y[x]]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\text {arctanh}\left (\frac {x}{\sqrt {x^2+4 e^{2 y(x)}}}\right )+y(x)=c_1,y(x)\right ]\\ \text {Solve}\left [2 y(x)-\log \left (\sqrt {x^2+4 e^{2 y(x)}}+x\right )=c_1,y(x)\right ]\\ y(x)&\to \frac {1}{2} \log \left (-\frac {x^2}{4}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*exp(-2*y(x))*Derivative(y(x), x) + 4*Derivative(y(x), x)**2 - exp(-2*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-x + sqrt(x**2 + 4*exp(2*y(x))))*exp(-2*y(x))/4 + Derivative(y(x), x) cannot be solved by the factorable group method