55.35.40 problem 40

Internal problem ID [14077]
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 : Friday, October 03, 2025 at 07:24:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\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. Time used: 2.592 (sec). Leaf size: 176

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]
 

\begin{align*} y(x)&\to c_1 \left (e^{\lambda x}\right )^{-\frac {k}{\lambda }}-\frac {b c_2 \lambda \left (e^{\lambda x}\right )^{\frac {b k-d}{b \lambda }} \left (a e^{\lambda x}+b\right )^{\frac {a d-b c}{a b \lambda }} \left (\frac {a e^{\lambda x}}{b}+1\right )^{\frac {b c-a d}{a b \lambda }} \operatorname {Hypergeometric2F1}\left (\frac {b c-a d}{a b \lambda },-\frac {d-2 b k}{b \lambda },\frac {b (2 k+\lambda )-d}{b \lambda },-\frac {a e^{x \lambda }}{b}\right )}{d-2 b k} \end{align*}

✗ 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