23.3.104 problem 106

Internal problem ID [5818]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 106
Date solved : Tuesday, September 30, 2025 at 02:03:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 99

ode=(b + c*E^x)*y[x] + a*D[y[x],x] + D[y[x],{x,2}] == 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-\sqrt {a^2-4 b}\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 b},2 \sqrt {c e^x}\right )+c_2 \operatorname {Gamma}\left (\sqrt {a^2-4 b}+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 b},2 \sqrt {c e^x}\right )\right ) \end{align*}

✗ Sympy

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

False