2.35.4 Problem 4

Internal problem ID [13928]
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 : 4
Date solved : Friday, December 19, 2025 at 08:50:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y&=0 \\ \end{align*}

2.35.4.1 ✓ Maple. Time used: 0.034 (sec). Leaf size: 57

ode:=diff(diff(y(x),x),x)-(a^2*exp(2*x)+a*(2*b+1)*exp(x)+b^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {WhittakerM}\left (-b , -b +\frac {1}{2}, 2 a \,{\mathrm e}^{x}\right ) c_2 +c_1 \,{\mathrm e}^{b x +a \,{\mathrm e}^{x}}-2 c_2 \,{\mathrm e}^{-a \,{\mathrm e}^{x}} \left (b -\frac {1}{2}\right ) \left (a \,{\mathrm e}^{x}\right )^{-b} 2^{-b} \]

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 
   <- Kovacics algorithm successful 
   Change of variables used: 
      [x = ln(t)] 
   Linear ODE actually solved: 
      (-a^2*t^2-2*a*b*t-a*t-b^2)*u(t)+t*diff(u(t),t)+t^2*diff(diff(u(t),t),t) =\ 
 0 
<- change of variables successful
 

2.35.4.2 ✓ Mathematica. Time used: 0.507 (sec). Leaf size: 57

ode=D[y[x],{x,2}]-(a^2*Exp[2*x]+a*(2*b+1)*Exp[x]+b^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{a e^x} \left (e^x\right )^{-b} \left (c_1 \left (e^x\right )^{2 b}-4^b c_2 \left (a e^x\right )^{2 b} \Gamma \left (-2 b,2 a e^x\right )\right ) \end{align*}

2.35.4.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq((-a**2*exp(2*x) - a*(2*b + 1)*exp(x) - b**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False