2.35.20 Problem 20

Internal problem ID [13944]
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 : 20
Date solved : Friday, December 19, 2025 at 08:51:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.35.20.1 ✓ Maple. Time used: 0.052 (sec). Leaf size: 135

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

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

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 
      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)/lambda] 
   Linear ODE actually solved: 
      (a*c*t+b*c-c^2)*u(t)+(a*lambda*t^2+b*lambda*t+lambda^2*t)*diff(u(t),t)+la\ 
mbda^2*t^2*diff(diff(u(t),t),t) = 0 
<- change of variables successful
 

2.35.20.2 ✓ Mathematica. Time used: 0.063 (sec). Leaf size: 96

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

\begin{align*} y(x)&\to (-1)^{-\frac {c}{\lambda }} c^{c/\lambda } \lambda ^{\frac {c}{\lambda }-1} a^{-\frac {c}{\lambda }} \left (c e^{\lambda x}\right )^{-\frac {c}{\lambda }} \left (c_2 (2 c-b) (-1)^{c/\lambda } \Gamma \left (-\frac {b-2 c}{\lambda },0,\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda \right ) \end{align*}

2.35.20.3 ✗ Sympy

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

False