2.35.25 Problem 25

Internal problem ID [13949]
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 : 25
Date solved : Friday, December 19, 2025 at 08:52:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.35.25.1 ✓ Maple. Time used: 0.069 (sec). Leaf size: 59

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

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

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

2.35.25.2 ✓ Mathematica. Time used: 0.532 (sec). Leaf size: 97

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

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

2.35.25.3 ✗ Sympy

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

False