81.17.2 problem 21-9

Internal problem ID [21735]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 21. Applications of second order differential equations
Problem number : 21-9
Date solved : Thursday, October 02, 2025 at 08:01:33 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} m y^{\prime \prime }+a y^{\prime }+k y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 44

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

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

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

ode=m*D[y[x],{x,2}]+a*D[y[x],x]+k*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.153 (sec). Leaf size: 46

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

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