62.16.2 problem Ex 2

Internal problem ID [12822]
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 2
Date solved : Monday, March 31, 2025 at 07:11:44 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 23

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

\[ y = \frac {\ln \left (2\right )}{2}-\frac {\ln \left (\frac {1}{2 \,{\mathrm e}^{2 c_1}+x}\right )}{2}+c_1 \]

✓ Mathematica. Time used: 0.365 (sec). Leaf size: 76

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

\begin{align*} y(x)\to \frac {1}{2} \log \left (\frac {1}{4} \left (e^{2 c_1}-2 e^{c_1} x\right )\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*Derivative(y(x), x) + 4*exp(2*y(x))*Derivative(y(x), x)**2 - 1,0) 
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