61.34.35 problem 35

Internal problem ID [12720]
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 : 35
Date solved : Monday, March 31, 2025 at 06:52:29 AM
CAS classification :

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

False