75.8.15 problem 213

Internal problem ID [16760]
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 : 213
Date solved : Monday, March 31, 2025 at 03:16:16 PM
CAS classification : [_quadrature]

\begin{align*} y&=\left (y^{\prime }-1\right ) {\mathrm e}^{y^{\prime }} \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 22

ode:=y(x) = (diff(y(x),x)-1)*exp(diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -1 \\ y &= \left (\ln \left (-c_1 +x \right )-1\right ) \left (-c_1 +x \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.218 (sec). Leaf size: 35

ode=y[x]==(D[y[x],x]-1)*Exp[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 {1}{W\left (\frac {K[1]}{e}\right )+1}dK[1]\&\right ][x+c_1] \\ y(x)\to -1 \\ \end{align*}

✓ Sympy. Time used: 0.449 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(Derivative(y(x), x) - 1)*exp(Derivative(y(x), x)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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