61.34.15 problem 15

Internal problem ID [12700]
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 : 15
Date solved : Friday, March 14, 2025 at 12:16:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 46

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

\[ y = \left (\left (\int {\mathrm e}^{\frac {-2 b \,{\mathrm e}^{\mu x} \lambda -{\mathrm e}^{\lambda x} a \mu }{\mu \lambda }}d x \right ) c_{1} +c_{2} \right ) {\mathrm e}^{\frac {b \,{\mathrm e}^{\mu x}}{\mu }} \]

✗ Mathematica

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

False