78.7.10 problem 2 (c)

Internal problem ID [18125]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 11 (Reduction of order). Problems at page 87
Problem number : 2 (c)
Date solved : Monday, March 31, 2025 at 05:12:29 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }&=y^{\prime } {\mathrm e}^{y} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.416 (sec). Leaf size: 15

ode:=diff(diff(y(x),x),x) = diff(y(x),x)*exp(y(x)); 
ic:=y(0) = 0, D(y)(0) = 2; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✗ Mathematica

ode=D[y[x],{x,2}]==D[y[x],x]*Exp[y[x]]; 
ic={y[0]==0,Derivative[1][y][0] == 2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

{}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - exp(-y(x))*Derivative(y(x), (x, 2)) cannot be solved by the factorable group method