61.34.34 problem 34

Internal problem ID [12719]
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 : 34
Date solved : Monday, March 31, 2025 at 06:52:26 AM
CAS classification :

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

False