2.35.11 Problem 11

Internal problem ID [13935]
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 : 11
Date solved : Friday, December 19, 2025 at 08:50:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y&=0 \\ \end{align*}

2.35.11.1 ✓ Maple. Time used: 0.026 (sec). Leaf size: 51

ode:=diff(diff(y(x),x),x)-diff(y(x),x)+(a*exp(3*lambda*x)+b*exp(2*lambda*x)+1/4-1/4*lambda^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {\left (\lambda -1\right ) x}{2}} \left (c_2 \operatorname {AiryBi}\left (-\frac {a \,{\mathrm e}^{\lambda x}+b}{\lambda ^{{2}/{3}} a^{{2}/{3}}}\right )+c_1 \operatorname {AiryAi}\left (-\frac {a \,{\mathrm e}^{\lambda x}+b}{\lambda ^{{2}/{3}} a^{{2}/{3}}}\right )\right ) \]

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 
   <- No Liouvillian solutions exists 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      <- Bessel successful 
   <- special function solution successful 
   Change of variables used: 
      [x = ln(t)/lambda] 
   Linear ODE actually solved: 
      (4*a*t^3+4*b*t^2-lambda^2+1)*u(t)+(4*lambda^2*t-4*lambda*t)*diff(u(t),t)+\ 
4*lambda^2*t^2*diff(diff(u(t),t),t) = 0 
<- change of variables successful
 

2.35.11.2 ✓ Mathematica. Time used: 0.37 (sec). Leaf size: 77

ode=D[y[x],{x,2}]-D[y[x],x]+(a*Exp[3*\[Lambda]*x]+b*Exp[2*\[Lambda]*x]+1/4-1/4*\[Lambda]^2  )*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{x/2} \left (c_1 \operatorname {AiryAi}\left (\frac {\left (e^{x \lambda } a+b\right ) \sqrt [3]{-\frac {a}{\lambda ^2}}}{a}\right )+c_2 \operatorname {AiryBi}\left (\frac {\left (e^{x \lambda } a+b\right ) \sqrt [3]{-\frac {a}{\lambda ^2}}}{a}\right )\right )}{\sqrt {e^{\lambda x}}} \end{align*}

2.35.11.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq((a*exp(3*lambda_*x) + b*exp(2*lambda_*x) - lambda_**2/4 + 1/4)*y(x) - 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')