90.14.14 problem 23

Internal problem ID [25242]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 3. Second Order Constant Coefficient Linear Differential Equations. Exercises at page 243
Problem number : 23
Date solved : Thursday, October 02, 2025 at 11:59:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+13 y&={\mathrm e}^{-3 t} \cos \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+13*y(t) = exp(-3*t)*cos(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-3 t} \left (\sin \left (2 t \right ) \left (t +4 c_2 \right )+4 \left (c_1 +\frac {1}{8}\right ) \cos \left (2 t \right )\right )}{4} \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 38

ode=D[y[t],{t,2}]+6*D[y[t],{t,1}]+13*y[t]==Exp[-3*t]*Cos[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{16} e^{-3 t} ((1+16 c_2) \cos (2 t)+4 (t+4 c_1) \sin (2 t)) \end{align*}

✓ Sympy. Time used: 0.257 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(13*y(t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-3*t)*cos(2*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{2} \cos {\left (2 t \right )} + \left (C_{1} + \frac {t}{4}\right ) \sin {\left (2 t \right )}\right ) e^{- 3 t} \]