60.7.58 problem 1666 (book 6.75)

Internal problem ID [11608]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1666 (book 6.75)
Date solved : Wednesday, March 05, 2025 at 02:34:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+x \,{\mathrm e}^{y}&=0 \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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