2.35.18 Problem 18

Internal problem ID [13942]
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 : 18
Date solved : Friday, December 19, 2025 at 08:51:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.35.18.1 ✓ Maple. Time used: 0.181 (sec). Leaf size: 79

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

\[ y = c_2 \sqrt {\pi }\, \left ({\mathrm e}^{\lambda x} a +{\mathrm e}^{-\lambda x} \lambda \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, {\mathrm e}^{\lambda x} \sqrt {a}}{2 \sqrt {\lambda }}\right )+\sqrt {a}\, \sqrt {\lambda }\, \sqrt {2}\, {\mathrm e}^{-\frac {a \,{\mathrm e}^{2 \lambda x}}{2 \lambda }} c_2 +c_1 \left ({\mathrm e}^{\lambda x} a +{\mathrm e}^{-\lambda x} \lambda \right ) \]

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 
   trying a quadrature 
   checking if the LODE has constant coefficients 
   checking if the LODE is of Euler type 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Trying a Liouvillian solution using Kovacics algorithm 
      A Liouvillian solution exists 
      Reducible group (found an exponential solution) 
      Group is reducible, not completely reducible 
      Solution has integrals. Trying a special function solution free of integr\ 
als... 
      -> Trying a solution in terms of special functions: 
         -> Bessel 
         -> elliptic 
         -> Legendre 
         <- Kummer successful 
      <- special function solution successful 
         -> Trying to convert hypergeometric functions to elementary form... 
         <- elementary form is not straightforward to achieve - returning speci\ 
al function solution free of uncomputed integrals 
      <- Kovacics algorithm successful 
   Change of variables used: 
      [x = ln(t)/lambda] 
   Linear ODE actually solved: 
      -t*a*u(t)+(a*t^2+2*lambda)*diff(u(t),t)+t*lambda*diff(diff(u(t),t),t) = 0 
<- change of variables successful
 

2.35.18.2 ✓ Mathematica. Time used: 0.06 (sec). Leaf size: 129

ode=D[y[x],{x,2}]+(a*Exp[2*\[Lambda]*x]+\[Lambda])*D[y[x],x]-a*\[Lambda]*Exp[2*\[Lambda]*x]*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt {2 \pi } c_2 \left (a e^{2 \lambda x}+\lambda \right ) \text {erf}\left (\frac {\sqrt {a \lambda e^{2 \lambda x}}}{\sqrt {2} \lambda }\right )-4 i \sqrt {2} a c_1 e^{2 \lambda x}+2 c_2 e^{-\frac {a e^{2 \lambda x}}{2 \lambda }} \sqrt {a \lambda e^{2 \lambda x}}-4 i \sqrt {2} c_1 \lambda }{4 \sqrt {a \lambda e^{2 \lambda x}}} \end{align*}

2.35.18.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-a*lambda_*y(x)*exp(2*lambda_*x) + (a*exp(2*lambda_*x) + lambda_)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_power_series_ordinary')