20.21.16 problem Problem 16

Internal problem ID [3943]
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 16
Date solved : Tuesday, March 04, 2025 at 05:19:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=10 \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 3.200 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+4*y(t) = 10*exp(-t); 
ic:=y(0) = 4, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 23

ode=D[y[t],{t,2}]+4*y[t]==10*Exp[-t]; 
ic={y[0]==4,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.080 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Derivative(y(t), (t, 2)) - 10*exp(-t),0) 
ics = {y(0): 4, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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