20.22.14 problem Problem 40

Internal problem ID [3969]
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 10.7. page 704
Problem number : Problem 40
Date solved : Sunday, March 30, 2025 at 02:13:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=5 \operatorname {Heaviside}\left (t -3\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: 0.458 (sec). Leaf size: 45

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = 5*Heaviside(t-3); 
ic:=y(0) = 2, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left (-\frac {1}{2}+i\right ) {\mathrm e}^{\left (-2+i\right ) \left (t -3\right )} \operatorname {Heaviside}\left (t -3\right )+\operatorname {Heaviside}\left (t -3\right )+\left (1-\frac {5 i}{2}\right ) {\mathrm e}^{\left (-2+i\right ) t}+\left (-\frac {1}{2}-i\right ) {\mathrm e}^{\left (-2-i\right ) \left (t -3\right )} \operatorname {Heaviside}\left (t -3\right )+\left (1+\frac {5 i}{2}\right ) {\mathrm e}^{\left (-2-i\right ) t} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 68

ode=D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==5*UnitStep[t-3]; 
ic={y[0]==2,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-2 t} (2 \cos (t)+5 \sin (t)) & t\leq 3 \\ e^{-2 t} \left (-e^6 \cos (3-t)+e^{2 t}+2 \cos (t)+2 e^6 \sin (3-t)+5 \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 2.124 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) - 5*Heaviside(t - 3) + 4*Derivative(y(t), 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)
 

\[ y{\left (t \right )} = \left (5 \sin {\left (t \right )} - 2 e^{6} \sin {\left (t - 3 \right )} \theta \left (t - 3\right ) + 2 \cos {\left (t \right )} - e^{6} \cos {\left (t - 3 \right )} \theta \left (t - 3\right )\right ) e^{- 2 t} + \theta \left (t - 3\right ) \]