30.11.7 problem 7

Internal problem ID [7587]
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 : 7
Date solved : Tuesday, September 30, 2025 at 04:54:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+4 y&=5 \sin \left (3 t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 41

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+4*y(t) = 5*sin(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t} \sin \left (\sqrt {3}\, t \right ) c_2 +{\mathrm e}^{-t} \cos \left (\sqrt {3}\, t \right ) c_1 -\frac {30 \cos \left (3 t \right )}{61}-\frac {25 \sin \left (3 t \right )}{61} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 55

ode=D[y[t],{t,2}]+2*D[y[t],t]+4*y[t]==5*Sin[3*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\frac {5}{61} (5 \sin (3 t)+6 \cos (3 t))+c_2 e^{-t} \cos \left (\sqrt {3} t\right )+c_1 e^{-t} \sin \left (\sqrt {3} t\right ) \end{align*}

✓ Sympy. Time used: 0.143 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 5*sin(3*t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (\sqrt {3} t \right )} + C_{2} \cos {\left (\sqrt {3} t \right )}\right ) e^{- t} - \frac {25 \sin {\left (3 t \right )}}{61} - \frac {30 \cos {\left (3 t \right )}}{61} \]