67.4.22 problem Problem 3(h)

Internal problem ID [14066]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 3(h)
Date solved : Tuesday, January 28, 2025 at 06:13:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=3 \operatorname {Heaviside}\left (t \right )-3 \operatorname {Heaviside}\left (t -4\right )+\left (2 t -5\right ) \operatorname {Heaviside}\left (t -4\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&={\frac {3}{4}}\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 9.488 (sec). Leaf size: 35

dsolve([diff(y(t),t$2)+4*y(t)=3*(Heaviside(t)-Heaviside(t-4))+(2*t-5)*Heaviside(t-4),y(0) = 3/4, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -4\right ) \sin \left (-8+2 t \right )}{4}+\frac {\operatorname {Heaviside}\left (t -4\right ) t}{2}+\sin \left (2 t \right )-2 \operatorname {Heaviside}\left (t -4\right )+\frac {3}{4} \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 60

DSolve[{D[y[t],{t,2}]+4*y[t]==3*(UnitStep[t]-UnitStep[t-4])+(2*t-5)*UnitStep[t-4],{y[0]==3/4,Derivative[1][y][0] ==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \sin (2 t)+\frac {3}{4} & 0\leq t\leq 4 \\ \frac {3}{4} \cos (2 t)+\sin (2 t) & t<0 \\ \frac {1}{4} (2 t+\sin (8-2 t)+4 \sin (2 t)-5) & \text {True} \\ \end {array} \\ \end {array} \]