23.3.53 problem 55

Internal problem ID [5767]
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 : 55
Date solved : Tuesday, September 30, 2025 at 02:02:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 39

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

\[ y = c_1 \operatorname {BesselJ}\left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {b x}{2}}}{b}\right )+c_2 \operatorname {BesselY}\left (0, \frac {2 \sqrt {a}\, {\mathrm e}^{\frac {b x}{2}}}{b}\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 55

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

\begin{align*} y(x)&\to c_1 \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {e^{b x}}}{b}\right )+2 c_2 \operatorname {BesselY}\left (0,\frac {2 \sqrt {a} \sqrt {e^{b x}}}{b}\right ) \end{align*}

✗ Sympy

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

False