46.6.9 problem 29

Internal problem ID [9630]
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. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 29
Date solved : Tuesday, September 30, 2025 at 06:21:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.126 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 95

ode=D[y[t],{t,2}]-y[t]==Exp[t]*Cos[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} \left (-e^{2 t} \int _1^0\frac {1}{2} \cos (K[1])dK[1]+e^{2 t} \int _1^t\frac {1}{2} \cos (K[1])dK[1]+\int _1^t-\frac {1}{2} e^{2 K[2]} \cos (K[2])dK[2]-\int _1^0-\frac {1}{2} e^{2 K[2]} \cos (K[2])dK[2]\right ) \end{align*}

✓ Sympy. Time used: 0.083 (sec). Leaf size: 24

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

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