30.11.9 problem 9

Internal problem ID [7589]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 4, Linear Second-Order Equations. EXERCISES 4.1 at page 156
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 04:54:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+4 y&=6 \cos \left (2 t \right )+8 \sin \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 41

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+4*y(t) = 6*cos(2*t)+8*sin(2*t); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 50

ode=D[y[t],{t,2}]+2*D[y[t],t]+4*y[t]==6*Cos[2*t]+8*Sin[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -2 \cos (2 t)+e^{-t} \left (3 e^t \sin (t) \cos (t)+c_2 \cos \left (\sqrt {3} t\right )+c_1 \sin \left (\sqrt {3} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.166 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 8*sin(2*t) - 6*cos(2*t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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