67.4.23 problem Problem 3(i)

Internal problem ID [14067]
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(i)
Date solved : Tuesday, January 28, 2025 at 06:13:30 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+4 y^{\prime }+5 y&=25 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 13.316 (sec). Leaf size: 86

dsolve([4*diff(y(t),t$2)+4*diff(y(t),t)+5*y(t)=25*t*(Heaviside(t)-Heaviside(t-Pi/2)),y(0) = 2, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = -4+\left (\frac {5}{4}-\frac {5 i}{8}\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \pi \,{\mathrm e}^{\left (\frac {1}{4}-\frac {i}{2}\right ) \left (-2 t +\pi \right )}+\left (\frac {5}{4}+\frac {5 i}{8}\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \pi \,{\mathrm e}^{\left (\frac {1}{4}+\frac {i}{2}\right ) \left (-2 t +\pi \right )}-3 \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )+\frac {4 \sin \left (t \right )}{3}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}+\left (4-5 t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )+6 \cos \left (t \right ) {\mathrm e}^{-\frac {t}{2}}+5 t \]

✓ Solution by Mathematica

Time used: 0.049 (sec). Leaf size: 101

DSolve[{4*D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==25*t*(UnitStep[t]-UnitStep[t-Pi/2]),{y[0]==2,Derivative[1][y][0] ==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 5 t+6 e^{-t/2} \cos (t)-4 & t\geq 0\land 2 t\leq \pi \\ e^{-t/2} (2 \cos (t)+3 \sin (t)) & t<0 \\ \frac {1}{4} e^{-t/2} \left (\left (24-e^{\pi /4} (12+5 \pi )\right ) \cos (t)+2 e^{\pi /4} (-8+5 \pi ) \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \]