61.34.31 problem 31

Internal problem ID [12716]
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 : 31
Date solved : Monday, March 31, 2025 at 06:52:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 81

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

\[ y = \left (c_2 \int \frac {{\mathrm e}^{\frac {-a \,{\mathrm e}^{x \left (\lambda +2 \mu \right )} \lambda -2 \left (-2 \mu x \lambda +{\mathrm e}^{\lambda x} b \right ) \left (\frac {\lambda }{2}+\mu \right )}{\lambda \left (\lambda +2 \mu \right )}}}{\left (a \,{\mathrm e}^{2 \mu x}+b \right )^{2}}d x +c_1 \right ) \left (a \,{\mathrm e}^{\mu x}+{\mathrm e}^{-\mu x} b \right ) \]

✗ Mathematica

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

False