23.3.106 problem 108

Internal problem ID [5820]
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 : 108
Date solved : Tuesday, September 30, 2025 at 02:03:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} b \,{\mathrm e}^{k x} y+a y^{\prime }+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 83

ode=b*E^(k*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-\frac {a}{k}\right ) \operatorname {BesselJ}\left (-\frac {a}{k},\frac {2 \sqrt {b e^{k x}}}{k}\right )+c_2 \operatorname {Gamma}\left (\frac {a+k}{k}\right ) \operatorname {BesselJ}\left (\frac {a}{k},\frac {2 \sqrt {b e^{k x}}}{k}\right )\right ) \end{align*}

✗ Sympy

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

False