61.34.40 problem 40

Internal problem ID [12725]
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 : 40
Date solved : Monday, March 31, 2025 at 06:53:23 AM
CAS classification :

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

✓ Maple. Time used: 0.468 (sec). Leaf size: 91

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

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

✗ Mathematica

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

✗ Sympy

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

False