20.21.20 problem Problem 20

Internal problem ID [3947]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.4. page 689
Problem number : Problem 20
Date solved : Tuesday, March 04, 2025 at 05:19:53 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=8 \sin \left (t \right )-6 \cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 2.742 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)-y(t) = 8*sin(t)-6*cos(t); 
ic:=y(0) = 2, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -4 \sin \left (t \right )+3 \sinh \left (t \right )-\cosh \left (t \right )+3 \cos \left (t \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 24

ode=D[y[t],{t,2}]-y[t]==8*Sin[t]-6*Cos[t]; 
ic={y[0]==2,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -2 e^{-t}+e^t-4 \sin (t)+3 \cos (t) \]

✓ Sympy. Time used: 0.081 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) - 8*sin(t) + 6*cos(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

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