61.4.23 problem Problem 3(i)

Internal problem ID [15348]
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 : Thursday, October 02, 2025 at 10:11:53 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*}

✓ Maple. Time used: 0.631 (sec). Leaf size: 75

ode:=4*diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = 25*t*(Heaviside(t)-Heaviside(t-1/2*Pi)); 
ic:=[y(0) = 2, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 101

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

\begin{align*} 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} \end{align*}

✓ Sympy. Time used: 2.253 (sec). Leaf size: 136

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-25*t*(Heaviside(t) - Heaviside(t - pi/2)) + 5*y(t) + 4*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 5 t \theta \left (t\right ) - 5 t \theta \left (t - \frac {\pi }{2}\right ) + \left (\left (- 3 \theta \left (t\right ) - 4 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right ) + \frac {5 \pi e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right )}{2} + 3\right ) \sin {\left (t \right )} + \left (4 \theta \left (t\right ) - \frac {5 \pi e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right )}{4} - 3 e^{\frac {\pi }{4}} \theta \left (t - \frac {\pi }{2}\right ) + 2\right ) \cos {\left (t \right )}\right ) e^{- \frac {t}{2}} - 4 \theta \left (t\right ) + 4 \theta \left (t - \frac {\pi }{2}\right ) \]