11.5.3 problem 3

Internal problem ID [1508]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number : 3
Date solved : Monday, January 27, 2025 at 04:58:26 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\delta \left (t -5\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: 0.360 (sec). Leaf size: 70

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=Dirac(t-5)+Heaviside(t-10),y(0) = 0, D(y)(0) = 1/2],y(t), singsol=all)
 

\[ y = -\frac {{\mathrm e}^{-2 t}}{2}+\frac {{\mathrm e}^{-t}}{2}+\frac {\operatorname {Heaviside}\left (t -10\right ) {\mathrm e}^{20-2 t}}{2}-\operatorname {Heaviside}\left (t -10\right ) {\mathrm e}^{10-t}+\frac {\operatorname {Heaviside}\left (t -10\right )}{2}-\operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{-2 t +10}+\operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{-t +5} \]

✓ Solution by Mathematica

Time used: 0.210 (sec). Leaf size: 71

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==DiracDelta[t-5]+UnitStep[t-10],{y[0]==0,Derivative[1][y][0] ==1/2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{2} e^{-2 t} \left (2 e^5 \left (e^t-e^5\right ) \theta (t-5)+\left (e^{10}-e^t\right )^2 (-\theta (10-t))+e^t+e^{2 t}-2 e^{t+10}+e^{20}-1\right ) \]