69.8.14 problem 212

Internal problem ID [18117]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 212
Date solved : Friday, October 03, 2025 at 07:32:21 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 62

ode:=y(x) = diff(y(x),x)*ln(diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 27

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {W(K[1])}{K[1]}dK[1]\&\right ][x+c_1] \end{align*}

✓ Sympy. Time used: 0.458 (sec). Leaf size: 29

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

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