30.11.8 problem 8

Internal problem ID [7588]
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 : 8
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 }+5 y&=-50 \sin \left (5 t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = -50*sin(5*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t} \sin \left (2 t \right ) c_2 +{\mathrm e}^{-t} \cos \left (2 t \right ) c_1 +2 \sin \left (5 t \right )+\cos \left (5 t \right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 40

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==-50*Sin[5*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 \sin (5 t)+\cos (5 t)+c_2 e^{-t} \cos (2 t)+c_1 e^{-t} \sin (2 t) \end{align*}

✓ Sympy. Time used: 0.138 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) + 50*sin(5*t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )}\right ) e^{- t} + 2 \sin {\left (5 t \right )} + \cos {\left (5 t \right )} \]