67.4.36 problem Problem 6(a)

Internal problem ID [14009]
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 6(a)
Date solved : Monday, March 31, 2025 at 08:21:45 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} 10 Q^{\prime }+100 Q&=\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} Q \left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.150 (sec). Leaf size: 41

ode:=10*diff(Q(t),t)+100*Q(t) = Heaviside(t-1)-Heaviside(t-2); 
ic:=Q(0) = 0; 
dsolve([ode,ic],Q(t),method='laplace');
 

\[ Q = -\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-10 t +10}}{100}+\frac {\operatorname {Heaviside}\left (t -1\right )}{100}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-10 t +20}}{100}-\frac {\operatorname {Heaviside}\left (t -2\right )}{100} \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 50

ode=10*D[ q[t],t]+100*q[t]==UnitStep[t-1]-UnitStep[t-2]; 
ic={q[0]==0}; 
DSolve[{ode,ic},q[t],t,IncludeSingularSolutions->True]
 

\[ q(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{100} e^{10-10 t} \left (-1+e^{10}\right ) & t>2 \\ \frac {1}{100} \left (1-e^{10-10 t}\right ) & 1<t\leq 2 \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.685 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
q = Function("q") 
ode = Eq(100*q(t) + Heaviside(t - 2) - Heaviside(t - 1) + 10*Derivative(q(t), t),0) 
ics = {q(0): 0} 
dsolve(ode,func=q(t),ics=ics)
 

\[ q{\left (t \right )} = - \frac {e^{10 - 10 t} \theta \left (t - 1\right )}{100} + \frac {e^{20 - 10 t} \theta \left (t - 2\right )}{100} - \frac {\theta \left (t - 2\right )}{100} + \frac {\theta \left (t - 1\right )}{100} \]