61.4.19 problem Problem 3(e)

Internal problem ID [15344]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 3(e)
Date solved : Thursday, October 02, 2025 at 10:11:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=5 \cos \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.361 (sec). Leaf size: 58

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

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 72

ode=D[y[t],{t,2}]+2*D[y[t],t]+2*y[t]==5*Cos[t]*(UnitStep[t]-UnitStep[t-Pi/2]); 
ic={y[0]==1,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} \cos (t) & t<0 \\ e^{-t} \left (\left (-3+2 e^{\pi /2}\right ) \sin (t)-e^{\pi /2} \cos (t)\right ) & 2 t>\pi \\ \cos (t)+\left (2-3 e^{-t}\right ) \sin (t) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 2.264 (sec). Leaf size: 99

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-5*Heaviside(t) + 5*Heaviside(t - pi/2))*cos(t) + 2*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- 3 \theta \left (t\right ) + 2 e^{\frac {\pi }{2}} \theta \left (t - \frac {\pi }{2}\right )\right ) \sin {\left (t \right )} + \left (- \theta \left (t\right ) - e^{\frac {\pi }{2}} \theta \left (t - \frac {\pi }{2}\right ) + 1\right ) \cos {\left (t \right )}\right ) e^{- t} + 2 \sin {\left (t \right )} \theta \left (t\right ) - 2 \sin {\left (t \right )} \theta \left (t - \frac {\pi }{2}\right ) + \cos {\left (t \right )} \theta \left (t\right ) - \cos {\left (t \right )} \theta \left (t - \frac {\pi }{2}\right ) \]