82.8.8 problem 36-9

Internal problem ID [21879]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-9
Date solved : Thursday, October 02, 2025 at 08:03:17 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 5

ode:=diff(diff(y(x),x),x) = diff(y(x),x)*exp(y(x)); 
ic:=[y(0) = -1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -1 \]

✗ Mathematica

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

{}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), (x, 2)) - exp(y(x))*Derivative(y(x), x),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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