61.34.39 problem 39

Internal problem ID [12724]
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 : 39
Date solved : Monday, March 31, 2025 at 06:52:41 AM
CAS classification :

\begin{align*} 2 \left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime \prime }+a \lambda \,{\mathrm e}^{\lambda x} y^{\prime }+c y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 109

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

\[ y = c_1 \sin \left (\frac {\sqrt {2}\, \sqrt {\frac {c}{a \,{\mathrm e}^{\lambda x}+b}}\, \sqrt {a \,{\mathrm e}^{\lambda x}+b}\, \operatorname {arctanh}\left (\frac {\sqrt {a \,{\mathrm e}^{\lambda x}+b}}{\sqrt {b}}\right )}{\lambda \sqrt {b}}\right )+c_2 \cos \left (\frac {\sqrt {2}\, \sqrt {\frac {c}{a \,{\mathrm e}^{\lambda x}+b}}\, \sqrt {a \,{\mathrm e}^{\lambda x}+b}\, \operatorname {arctanh}\left (\frac {\sqrt {a \,{\mathrm e}^{\lambda x}+b}}{\sqrt {b}}\right )}{\lambda \sqrt {b}}\right ) \]

✗ Mathematica

ode=2*(a*Exp[\[Lambda]*x]+b)*D[y[x],{x,2}]+a*\[Lambda]*Exp[\[Lambda]*x]*D[y[x],x]+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") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(a*lambda_*exp(lambda_*x)*Derivative(y(x), x) + c*y(x) + (2*a*exp(lambda_*x) + 2*b)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False