61.34.36 problem 36

Internal problem ID [12721]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.3-1. Equations with exponential functions
Problem number : 36
Date solved : Monday, March 31, 2025 at 06:52:31 AM
CAS classification :

\begin{align*} y^{\prime \prime }+a \,{\mathrm e}^{b \,x^{n}} y^{\prime }+c \left (a \,{\mathrm e}^{b \,x^{n}}-c \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

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

\[ y = \left (\int {\mathrm e}^{2 c x -a \int {\mathrm e}^{b \,x^{n}}d x}d x c_1 +c_2 \right ) {\mathrm e}^{-c x} \]

✗ Mathematica

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

✗ Sympy

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

False