82.17.4 problem Ex. 4

Internal problem ID [18749]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 37
Problem number : Ex. 4
Date solved : Monday, March 31, 2025 at 06:07:37 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} {\mathrm e}^{4 x} \left (y^{\prime }-1\right )+{\mathrm e}^{2 y} {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.343 (sec). Leaf size: 59

ode:=exp(4*x)*(diff(y(x),x)-1)+exp(2*y(x))*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {arctanh}\left (\operatorname {RootOf}\left (-1+\left (4 \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{16+4 c_1 -4 x}+2 \,{\mathrm e}^{8+\textit {\_Z}}+{\mathrm e}^{16}-{\mathrm e}^{2 \textit {\_Z} -4 c_1 +4 x}\right )}+{\mathrm e}^{8}\right ) \textit {\_Z}^{2}\right ) {\mathrm e}^{4}\right )+2 c_1 \]

✓ Mathematica. Time used: 0.958 (sec). Leaf size: 197

ode=Exp[4*x]*(D[y[x],x]-1)+Exp[2*y[x]]*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\frac {y(x)}{2}-\frac {e^{-2 x} \sqrt {4 e^{2 y(x)+4 x}+e^{8 x}} \text {arctanh}\left (\frac {e^{2 x}}{\sqrt {4 e^{2 y(x)}+e^{4 x}}}\right )}{2 \sqrt {4 e^{2 y(x)}+e^{4 x}}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {e^{-2 x} \sqrt {4 e^{2 y(x)+4 x}+e^{8 x}} \text {arctanh}\left (\frac {e^{2 x}}{\sqrt {4 e^{2 y(x)}+e^{4 x}}}\right )}{2 \sqrt {4 e^{2 y(x)}+e^{4 x}}}+\frac {y(x)}{2}&=c_1,y(x)\right ] \\ y(x)\to \frac {1}{2} \left (\log \left (-\frac {e^{8 x}}{4}\right )-4 x\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((Derivative(y(x), x) - 1)*exp(4*x) + exp(2*y(x))*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out