20.22.10 problem Problem 36

Internal problem ID [3965]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.7. page 704
Problem number : Problem 36
Date solved : Monday, January 27, 2025 at 08:05:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y&=\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=4 \end{align*}

✓ Solution by Maple

Time used: 3.496 (sec). Leaf size: 35

dsolve([diff(y(t),t$2)-4*y(t)=Heaviside(t-1)-Heaviside(t-2),y(0) = 0, D(y)(0) = 4],y(t), singsol=all)
 

\[ y = \frac {\operatorname {Heaviside}\left (-1+t \right ) \sinh \left (-1+t \right )^{2}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right ) \sinh \left (t -2\right )^{2}}{2}+2 \sinh \left (2 t \right ) \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 113

DSolve[{D[y[t],{t,2}]-4*y[t]==UnitStep[t-1]-UnitStep[t-2],{y[0]==0,Derivative[1][y][0] ==4}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-2 t} \left (-1+e^{4 t}\right ) & t\leq 1 \\ \frac {1}{8} \left (-2+e^{2-2 t}-8 e^{-2 t}+8 e^{2 t}+e^{2 t-2}\right ) & 1<t\leq 2 \\ \frac {1}{8} e^{-2 (t+2)} \left (-8 e^4+e^6-e^8-e^{4 t}+e^{4 t+2}+8 e^{4 t+4}\right ) & \text {True} \\ \end {array} \\ \end {array} \]