76.21.3 problem 3

Internal problem ID [17767]
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 : Tuesday, January 28, 2025 at 11:02:40 AM
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*}

✓ Solution by Maple

Time used: 13.195 (sec). Leaf size: 76

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=Dirac(t-Pi)+Heaviside(t-10),y(0) = 0, D(y)(0) = 1/2],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.241 (sec). Leaf size: 73

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

\[ y(t)\to \frac {1}{2} e^{-2 t} \left (-2 e^{\pi } \left (e^{\pi }-e^t\right ) \theta (t-\pi )+\left (e^{10}-e^t\right )^2 (-\theta (10-t))+e^t+e^{2 t}-2 e^{t+10}+e^{20}-1\right ) \]