2.35.31 Problem 31

Internal problem ID [13955]
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 : Friday, December 19, 2025 at 08:52:58 PM
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*}

2.35.31.1 ✓ Maple. Time used: 0.014 (sec). Leaf size: 81

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

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

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x),\ 
 dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
<- unable to find a useful change of variables 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   trying 2nd order exact linear 
   trying symmetries linear in x and y(x) 
   trying to convert to a linear ODE with constant coefficients 
   trying 2nd order, integrating factor of the form mu(x,y) 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a pow\ 
er @ Moebius 
   -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(\ 
x), dx)) * 2F1([a1, a2], [b1], f) 
   -> Trying changes of variables to rationalize or make the ODE simpler 
   <- unable to find a useful change of variables 
      trying a symmetry of the form [xi=0, eta=F(x)] 
   trying to convert to an ODE of Bessel type 
   -> trying reduction of order to Riccati 
      trying Riccati sub-methods: 
         trying Riccati_symmetries 
         -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
         -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
         -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
--- Trying Lie symmetry methods, 2nd order --- 
   -> Computing symmetries using: way = 3 
[0, y] 
   <- successful computation of symmetries. 
   -> Computing symmetries using: way = 5 
[0, y], [0, a*exp(mu*x)+exp(-mu*x)*b] 
   <- successful computation of symmetries.
 

2.35.31.2 ✗ 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

2.35.31.3 ✗ 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