61.34.33 problem 33

Internal problem ID [12718]
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 : 33
Date solved : Monday, March 31, 2025 at 06:52:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

ode:=diff(diff(y(x),x),x)+(exp(lambda*x)*a+b*exp(x*mu)+c)*diff(y(x),x)+(a*b*exp(x*(lambda+mu))+exp(lambda*x)*a*c+b*mu*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]+b*Exp[\[Mu]*x]+c)*D[y[x],x]+(a*b*Exp[(\[Lambda]+\[Mu])*x]+a*c*Exp[\[Lambda]*x]+b*\[Mu]*Exp[\[Mu]*x])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ 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) + b*exp(mu*x) + c)*Derivative(y(x), x) + (a*b*exp(x*(lambda_ + mu)) + a*c*exp(lambda_*x) + b*mu*exp(mu*x))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False