38.1.50 problem 56

Internal problem ID [8211]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Exercises 1.1 at page 12
Problem number : 56
Date solved : Tuesday, September 30, 2025 at 05:18:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 37

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

\[ y = {\mathrm e}^{-t} \sin \left (\sqrt {3}\, t \right ) c_2 +{\mathrm e}^{-t} \cos \left (\sqrt {3}\, t \right ) c_1 +\frac {15 \sin \left (t \right )}{13}-\frac {10 \cos \left (t \right )}{13} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 51

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

\begin{align*} y(t)&\to -\frac {5}{13} (2 \cos (t)-3 \sin (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: 0.132 (sec). Leaf size: 39

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

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (\sqrt {3} t \right )} + C_{2} \cos {\left (\sqrt {3} t \right )}\right ) e^{- t} + \frac {15 \sin {\left (t \right )}}{13} - \frac {10 \cos {\left (t \right )}}{13} \]