2.35.22 Problem 22

Internal problem ID [13946]
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 : 22
Date solved : Friday, December 19, 2025 at 08:52:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.35.22.1 ✓ Maple. Time used: 0.123 (sec). Leaf size: 205

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

\[ y = {\mathrm e}^{-\frac {{\mathrm e}^{\lambda x} \left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right )}{2 \lambda }} \left (\operatorname {KummerU}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (a +b -2 \lambda \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {a +b -2 \lambda }{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_2 +\operatorname {KummerM}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (a +b -2 \lambda \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {a +b -2 \lambda }{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_1 \right ) \]

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 
      -> elliptic 
      -> Legendre 
      -> Kummer 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
         <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
      <- Kummer successful 
   <- special function solution successful 
   Change of variables used: 
      [x = ln(t)/lambda] 
   Linear ODE actually solved: 
      (a^2*b*t-a^2*lambda*t)*u(t)+(b*t+a+b-2*lambda)*diff(u(t),t)+t*lambda*diff\ 
(diff(u(t),t),t) = 0 
<- change of variables successful
 

2.35.22.2 ✓ Mathematica. Time used: 1.227 (sec). Leaf size: 260

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

\begin{align*} y(x)&\to \exp \left (-\frac {e^{\lambda x} \left (\sqrt {-4 a^2 b \lambda +4 a^2 \lambda ^2+b^2}+b\right )}{2 \lambda }\right ) \left (c_1 \operatorname {HypergeometricU}\left (\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}},\frac {a+b-2 \lambda }{\lambda },\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )+c_2 L_{-\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}}^{\frac {a+b-3 \lambda }{\lambda }}\left (\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )\right ) \end{align*}

2.35.22.3 ✗ Sympy

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