61.4.22 problem Problem 3(h)

Internal problem ID [15347]
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 : Thursday, October 02, 2025 at 10:11:52 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.282 (sec). Leaf size: 35

ode:=diff(diff(y(t),t),t)+4*y(t) = 3*Heaviside(t)-3*Heaviside(t-4)+(2*t-5)*Heaviside(t-4); 
ic:=[y(0) = 3/4, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 60

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

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

✓ Sympy. Time used: 1.646 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((5 - 2*t)*Heaviside(t - 4) + 4*y(t) - 3*Heaviside(t) + 3*Heaviside(t - 4) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3/4, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

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