62.16.3 problem Ex 3

Internal problem ID [12823]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 27. Clairaut equation. Page 56
Problem number : Ex 3
Date solved : Monday, March 31, 2025 at 07:11:54 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.376 (sec). Leaf size: 57

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

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

✓ Mathematica. Time used: 0.894 (sec). Leaf size: 176

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

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

✗ Sympy

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

Timed Out