70.21.11 problem 11

Internal problem ID [19061]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 11
Date solved : Thursday, October 02, 2025 at 03:37:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=\cos \left (t \right )+\delta \left (t -\frac {\pi }{2}\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: 0.458 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 0.155 (sec). Leaf size: 141

ode=D[y[t],{t,2}]+2*D[y[t],t]+2*y[t]==Cos[t]+DiracDelta[t-Pi/2]; 
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 (\sin (t) \int _1^0e^{K[1]} \cos (K[1]) (\cos (K[1])+2 \delta (\pi -2 K[1]))dK[1]-\sin (t) \int _1^te^{K[1]} \cos (K[1]) (\cos (K[1])+2 \delta (\pi -2 K[1]))dK[1]+\cos (t) \int _1^0-e^{K[2]} (\cos (K[2])+2 \delta (\pi -2 K[2])) \sin (K[2])dK[2]-\cos (t) \int _1^t-e^{K[2]} (\cos (K[2])+2 \delta (\pi -2 K[2])) \sin (K[2])dK[2]\right ) \end{align*}

✓ Sympy. Time used: 5.782 (sec). Leaf size: 102

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi/2) + 2*y(t) - cos(t) + 2*Derivative(y(t), 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)
 

\[ y{\left (t \right )} = \left (\left (- \int \limits ^{0} e^{t} \cos ^{2}{\left (t \right )}\, dt + \int \left (\operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} + \cos {\left (t \right )}\right ) e^{t} \cos {\left (t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} e^{t} \cos {\left (t \right )}\, dt\right ) \sin {\left (t \right )} + \left (- \int \left (\operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} + \cos {\left (t \right )}\right ) e^{t} \sin {\left (t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} e^{t} \sin {\left (t \right )}\, dt + \int \limits ^{0} e^{t} \sin {\left (t \right )} \cos {\left (t \right )}\, dt\right ) \cos {\left (t \right )}\right ) e^{- t} \]