62.2.6 problem Problem 3.7(f)

Internal problem ID [15432]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 3 Bessel functions. Problems page 89
Problem number : Problem 3.7(f)
Date solved : Thursday, October 02, 2025 at 10:14:05 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\beta y^{\prime }+\gamma y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)+beta*diff(y(x),x)+gamma*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 \,{\mathrm e}^{x \sqrt {\beta ^{2}-4 \gamma }}+c_2 \right ) {\mathrm e}^{-\frac {\left (\beta +\sqrt {\beta ^{2}-4 \gamma }\right ) x}{2}} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 47

ode=D[y[x],{x,2}]+\[Beta]*D[y[x],x]+\[Gamma]*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-\frac {1}{2} x \left (\sqrt {\beta ^2-4 \gamma }+\beta \right )} \left (c_2 e^{x \sqrt {\beta ^2-4 \gamma }}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} e^{\frac {x \left (- \beta + \sqrt {\beta ^{2} - 4 \Gamma }\right )}{2}} + C_{2} e^{- \frac {x \left (\beta + \sqrt {\beta ^{2} - 4 \Gamma }\right )}{2}} \]