74.10.37 problem 39

Internal problem ID [16171]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.2, page 147
Problem number : 39
Date solved : Monday, March 31, 2025 at 02:45:54 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} a y^{\prime \prime }+2 b y^{\prime }+c y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

ode:=a*diff(diff(y(t),t),t)+2*b*diff(y(t),t)+c*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 54

ode=a*D[y[t],{t,2}]+2*b*D[y[t],t]+c*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-\frac {t \left (\sqrt {b^2-a c}+b\right )}{a}} \left (c_2 e^{\frac {2 t \sqrt {b^2-a c}}{a}}+c_1\right ) \]

✓ Sympy. Time used: 0.224 (sec). Leaf size: 39

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

\[ y{\left (t \right )} = C_{1} e^{\frac {t \left (- b + \sqrt {- a c + b^{2}}\right )}{a}} + C_{2} e^{- \frac {t \left (b + \sqrt {- a c + b^{2}}\right )}{a}} \]