46.5.11 problem 41

Internal problem ID [9620]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.2.2 TRANSFORMS OF DERIVATIVES Page 289
Problem number : 41
Date solved : Tuesday, September 30, 2025 at 06:21:33 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.125 (sec). Leaf size: 25

ode:=diff(y(t),t)+y(t) = exp(-3*t)*cos(2*t); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 30

ode=D[y[t],t]+y[t]==Exp[-3*t]*Cos[2*t]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} \int _0^te^{-2 K[1]} \cos (2 K[1])dK[1] \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 26

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

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