60.7.59 problem 1667 (book 6.76)

Internal problem ID [11609]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1667 (book 6.76)
Date solved : Wednesday, March 05, 2025 at 02:34:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=b*E^y[x]*x + a*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") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*Derivative(y(x), x) + b*x*exp(y(x)) + x*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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