70.21.3 problem 3

Internal problem ID [19053]
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 : 3
Date solved : Thursday, October 02, 2025 at 03:37:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\delta \left (t -\pi \right )+\operatorname {Heaviside}\left (t -10\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.217 (sec). Leaf size: 76

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = Dirac(t-Pi)+Heaviside(t-10); 
ic:=[y(0) = 0, D(y)(0) = 1/2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-t}}{2}-\frac {{\mathrm e}^{-2 t}}{2}+\frac {\operatorname {Heaviside}\left (t -10\right ) {\mathrm e}^{20-2 t}}{2}-\operatorname {Heaviside}\left (t -10\right ) {\mathrm e}^{-t +10}+\frac {\operatorname {Heaviside}\left (t -10\right )}{2}+\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-t +\pi }-\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 \pi -2 t} \]

✓ Mathematica. Time used: 2.004 (sec). Leaf size: 394

ode=D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==DiracDelta[t-Pi]+UnitStep[t-10]; 
ic={y[0]==0,Derivative[1][y][0] ==1/2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \theta (10-t) \left (\frac {1}{2} e^{-2 t} \left (2 e^t \int _1^0e^{\pi } \delta (\pi -K[2])dK[2]-2 \int _1^t-e^{2 K[3]} (\delta (K[3]-\pi )+1)dK[3]-2 e^t \int _1^te^{K[4]} (\delta (K[4]-\pi )+1)dK[4]+2 \int _1^0-e^{2 \pi } \delta (\pi -K[1])dK[1]-e^t-2 e^{t+1}+2 e^{t+10}-e^{20}+e^2+1\right )-\frac {1}{2} e^{-2 t} \left (-2 \int _1^t-e^{2 \pi } \delta (\pi -K[1])dK[1]+2 e^t \int _1^0e^{\pi } \delta (\pi -K[2])dK[2]-2 e^t \int _1^te^{\pi } \delta (\pi -K[2])dK[2]+2 \int _1^0-e^{2 \pi } \delta (\pi -K[1])dK[1]-e^t+1\right )\right )-\frac {1}{2} e^{-2 t} \left (2 e^t \int _1^0e^{\pi } \delta (\pi -K[2])dK[2]-2 \int _1^t-e^{2 K[3]} (\delta (K[3]-\pi )+1)dK[3]-2 e^t \int _1^te^{K[4]} (\delta (K[4]-\pi )+1)dK[4]+2 \int _1^0-e^{2 \pi } \delta (\pi -K[1])dK[1]-e^t-2 e^{t+1}+2 e^{t+10}-e^{20}+e^2+1\right ) \end{align*}

✓ Sympy. Time used: 0.964 (sec). Leaf size: 99

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi) + 2*y(t) - Heaviside(t - 10) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1/2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \int \left (\operatorname {Dirac}{\left (t - \pi \right )} + \theta \left (t - 10\right )\right ) e^{2 t}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} e^{2 t}\, dt + \int \limits ^{0} e^{2 t} \theta \left (t - 10\right )\, dt - \frac {1}{2}\right ) e^{- t} + \int \left (\operatorname {Dirac}{\left (t - \pi \right )} + \theta \left (t - 10\right )\right ) e^{t}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} e^{t}\, dt - \int \limits ^{0} e^{t} \theta \left (t - 10\right )\, dt + \frac {1}{2}\right ) e^{- t} \]