2.35.12 Problem 12

Internal problem ID [13936]
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 : 12
Date solved : Friday, December 19, 2025 at 08:50:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.35.12.1 ✓ Maple. Time used: 0.103 (sec). Leaf size: 214

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

\[ y = \frac {\left (-\pi c_1 {\left (-\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{2+n}}{\lambda ^{2} b^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{2 n +4}} \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{2+n}}{\lambda ^{2} b^{2} \left (2+n \right )^{2}}}\right ) \csc \left (\frac {\pi \left (n +3\right )}{2+n}\right )+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{2+n}}{\lambda ^{2} b^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{2 n +4}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{2+n}}{\lambda ^{2} b^{2} \left (2+n \right )^{2}}}\right ) c_2 \left (2+n \right ) \left (b \,{\mathrm e}^{\lambda x}+c \right )\right ) {\mathrm e}^{-\frac {\left (\lambda -1\right ) x}{2}}}{\left (2+n \right ) \Gamma \left (\frac {n +3}{2+n}\right )} \]

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 symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Trying an equivalence, under non-integer power transformations, 
      to LODEs admitting Liouvillian solutions. 
      -> Trying a Liouvillian solution using Kovacics algorithm 
      <- No Liouvillian solutions exists 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Whittaker 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
         <- hyper3 successful: received ODE is equivalent to the 0F1 ODE 
      <- Whittaker successful 
   <- special function solution successful 
   Change of variables used: 
      [x = ln(t)/lambda] 
   Linear ODE actually solved: 
      (4*a*t^2*(b*t+c)^n-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
 

2.35.12.2 ✗ Mathematica

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

Not solved

2.35.12.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq((a*(b*exp(lambda_*x) + c)**n*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