61.34.37 problem 37

Internal problem ID [12722]
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 : 37
Date solved : Monday, March 31, 2025 at 06:52:33 AM
CAS classification :

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 52

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

\[ y = -c_1 \left (a \,{\mathrm e}^{\lambda x}+b \right ) \ln \left (a \,{\mathrm e}^{\lambda x}+b \right )+c_1 \left (a \,{\mathrm e}^{\lambda x}+b \right ) \ln \left ({\mathrm e}^{\lambda x}\right )+{\mathrm e}^{\lambda x} c_2 a +b \left (c_1 +c_2 \right ) \]

✗ Mathematica

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

False