81.8.5 problem 9-5

Internal problem ID [21588]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 9. Clairaut Equation. Page 133.
Problem number : 9-5
Date solved : Friday, October 03, 2025 at 07:58:58 AM
CAS classification : [_quadrature]

\begin{align*} 2 y^{\prime }+y-2 y^{\prime } \ln \left (y^{\prime }\right )&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 51

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

\begin{align*} y &= -2 \,{\mathrm e}^{-\sqrt {1-c_1 +x}} \left (1+\sqrt {1-c_1 +x}\right ) \\ y &= 2 \,{\mathrm e}^{\sqrt {1-c_1 +x}} \left (-1+\sqrt {1-c_1 +x}\right ) \\ \end{align*}

✓ Mathematica. Time used: 1.877 (sec). Leaf size: 72

ode=2*D[y[x],x]+y[x]-2*D[y[x],x]*Log[D[y[x],x]]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -2 e^{-\sqrt {x+1+2 c_1}} \left (1+\sqrt {x+1+2 c_1}\right )\\ y(x)&\to 2 e^{\sqrt {x+1+2 c_1}} \left (-1+\sqrt {x+1+2 c_1}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.614 (sec). Leaf size: 48

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

\[ - e x + \frac {y{\left (x \right )} e^{- W\left (\frac {y{\left (x \right )}}{2 e}\right )} W\left (\frac {y{\left (x \right )}}{2 e}\right )}{2} + y{\left (x \right )} e^{- W\left (\frac {y{\left (x \right )}}{2 e}\right )} = C_{1} \]