23.2.90 problem 92

Internal problem ID [5445]
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 : 92
Date solved : Tuesday, September 30, 2025 at 12:43:27 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.596 (sec). Leaf size: 157

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

\begin{align*} y &= c_1 -\operatorname {arctanh}\left (\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (6 \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 x +2 \textit {\_Z} -2 c_1}-4 \,{\mathrm e}^{\textit {\_Z} +2 c_1 -2 x}+1-{\mathrm e}^{2 c_1 -2 x}-4 \,{\mathrm e}^{3 \textit {\_Z} +2 x -2 c_1}+8 \,{\mathrm e}^{2 \textit {\_Z}}\right )}-1\right )}\right ) \\ y &= c_1 +\operatorname {arctanh}\left (\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{2}-4 \,{\mathrm e}^{\operatorname {RootOf}\left (6 \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 x +2 \textit {\_Z} -2 c_1}-4 \,{\mathrm e}^{\textit {\_Z} +2 c_1 -2 x}+1-{\mathrm e}^{2 c_1 -2 x}-4 \,{\mathrm e}^{3 \textit {\_Z} +2 x -2 c_1}+8 \,{\mathrm e}^{2 \textit {\_Z}}\right )}-1\right )}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.557 (sec). Leaf size: 176

ode=4 (D[y[x],x])^2+2 Exp[2 x-2 y[x]] D[y[x],x]-Exp[2 x-2 y[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 - 2*y(x))*Derivative(y(x), x) - exp(2*x - 2*y(x)) + 4*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out