54.7.66 problem 1674 (book 6.83)

Internal problem ID [12915]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1674 (book 6.83)
Date solved : Wednesday, October 01, 2025 at 02:46:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+a \left ({\mathrm e}^{y}-1\right )&=0 \end{align*}

✗ Maple

ode:=x^2*diff(diff(y(x),x),x)+a*(exp(y(x))-1) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=a*(-1 + E^y[x]) + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

NotImplementedError : solve: Cannot solve a*(exp(y(x)) - 1) + x**2*Derivative(y(x), (x, 2))