55.35.10 problem 10

Internal problem ID [14047]
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 : 10
Date solved : Thursday, October 02, 2025 at 09:09:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.065 (sec). Leaf size: 69

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

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

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 123

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

\begin{align*} y(x)&\to e^{-\frac {a x}{2}} \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^2-4 c}}{\lambda }\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^2-4 c}}{\lambda },\frac {2 \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_2 \operatorname {Gamma}\left (\frac {\lambda +\sqrt {a^2-4 c}}{\lambda }\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^2-4 c}}{\lambda },\frac {2 \sqrt {b e^{x \lambda }}}{\lambda }\right )\right ) \end{align*}

✗ Sympy

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

False