2.35.16 Problem 16

Internal problem ID [13940]
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 : 16
Date solved : Friday, December 19, 2025 at 08:51:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.35.16.1 ✗ Maple

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

\[ \text {No solution found} \]

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 
<- unable to find a useful change of variables 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   trying 2nd order exact linear 
   trying symmetries linear in x and y(x) 
   trying to convert to a linear ODE with constant coefficients 
   trying 2nd order, integrating factor of the form mu(x,y) 
   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 pow\ 
er @ 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 
   <- unable to find a useful change of variables 
      trying a symmetry of the form [xi=0, eta=F(x)] 
   trying to convert to an ODE of Bessel type 
   -> trying reduction of order to Riccati 
      trying Riccati sub-methods: 
         trying Riccati_symmetries 
         -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
         -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
         -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
--- Trying Lie symmetry methods, 2nd order --- 
   -> Computing symmetries using: way = 3 
[0, y] 
   <- successful computation of symmetries. 
   -> Computing symmetries using: way = 5
 

2.35.16.2 ✓ Mathematica. Time used: 1.269 (sec). Leaf size: 232

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

\begin{align*} y(x)&\to 2^{\frac {1}{2}-\frac {i \sqrt {c}}{\lambda }} \left (e^x\right )^{\frac {1}{2}-\frac {\lambda }{2}} \left (\left (e^x\right )^{\mu }\right )^{\left .-\frac {1}{2}\right /\mu } \left (\left (e^x\right )^{\lambda }\right )^{\frac {1}{2}-\frac {i \sqrt {c}}{\lambda }} e^{-\frac {k \left (e^x\right )^{\mu }}{\mu }+\frac {i \sqrt {a} \left (e^x\right )^{\lambda }}{\lambda }} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {\frac {i b}{\sqrt {a}}-\lambda +2 i \sqrt {c}}{2 \lambda },1-\frac {2 i \sqrt {c}}{\lambda },-\frac {2 i \sqrt {a} \left (e^x\right )^{\lambda }}{\lambda }\right )+c_2 L_{\frac {\frac {i b}{\sqrt {a}}-\lambda +2 i \sqrt {c}}{2 \lambda }}^{-\frac {2 i \sqrt {c}}{\lambda }}\left (-\frac {2 i \sqrt {a} \left (e^x\right )^{\lambda }}{\lambda }\right )\right ) \end{align*}

2.35.16.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
k = symbols("k") 
lambda_ = symbols("lambda_") 
mu = symbols("mu") 
y = Function("y") 
ode = Eq(2*k*exp(mu*x)*Derivative(y(x), x) + (a*exp(2*lambda_*x) + b*exp(lambda_*x) + c + k**2*exp(2*mu*x) + k*mu*exp(mu*x))*y(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')