61.34.41 problem 41

Internal problem ID [12726]
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 : 41
Date solved : Monday, March 31, 2025 at 06:53:26 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 }+\left (n \,{\mathrm e}^{\lambda x}+m \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.295 (sec). Leaf size: 296

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

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

✗ Mathematica

ode=(a*Exp[\[Lambda]*x]+b)*D[y[x],{x,2}]+(c*Exp[\[Lambda]*x]+d)*D[y[x],x]+(n*Exp[\[Lambda]*x]+m)*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") 
n = symbols("n") 
m = symbols("m") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq((m + n*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