2.35.26 Problem 26

Internal problem ID [13950]
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 : 26
Date solved : Friday, December 19, 2025 at 08:52:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.35.26.1 ✓ Maple. Time used: 0.051 (sec). Leaf size: 88

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

\[ y = c_1 \,{\mathrm e}^{-k x -{\mathrm e}^{x} c}+c_2 \left (-2 c +a \right ) \left (-{\mathrm e}^{{\mathrm e}^{x} \left (-\frac {a}{2}+c \right )} \left (-1+b -2 k \right ) \left (\left (-2 c +a \right ) {\mathrm e}^{x}\right )^{-\frac {b}{2}+k}+\operatorname {WhittakerM}\left (-\frac {b}{2}+k , -\frac {b}{2}+k +\frac {1}{2}, \left (-2 c +a \right ) {\mathrm e}^{x}\right )\right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{x}}{2}-\frac {b x}{2}} \]

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

2.35.26.2 ✓ Mathematica. Time used: 1.038 (sec). Leaf size: 71

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

\begin{align*} y(x)&\to e^{-c e^x} \left (e^x\right )^{-k} \left (c_1-c_2 \left (e^x\right )^{2 k-b} \left (e^x (a-2 c)\right )^{b-2 k} \Gamma \left (2 k-b,(a-2 c) e^x\right )\right ) \end{align*}

2.35.26.3 ✗ Sympy

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

False