70.22.4 problem 17

Internal problem ID [19071]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 17
Date solved : Thursday, October 02, 2025 at 03:37:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=1-\operatorname {Heaviside}\left (t -\pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.320 (sec). Leaf size: 52

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

\[ y = -\frac {4 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{2}\right ) \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{2}}}{5}-\frac {4 \operatorname {Heaviside}\left (t -\pi \right )}{5}+\frac {4}{5}+\frac {{\mathrm e}^{-\frac {t}{2}} \left (2 \cos \left (t \right )-9 \sin \left (t \right )\right )}{10} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 80

ode=D[y[t],{t,2}]+D[y[t],t]+125/100*y[t]==1-UnitStep[t-Pi]; 
ic={y[0]==1,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{10} \left (2 e^{-t/2} \cos (t)-9 e^{-t/2} \sin (t)+8\right ) & t\leq \pi \\ \frac {1}{10} e^{-t/2} \left (\left (2-8 e^{\pi /2}\right ) \cos (t)-\left (9+4 e^{\pi /2}\right ) \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 1.150 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t)/4 + Heaviside(t - pi) + Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 1,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \frac {4 e^{\frac {\pi }{2}} \theta \left (t - \pi \right )}{5} + \frac {1}{5}\right ) \cos {\left (t \right )} + \left (- \frac {2 e^{\frac {\pi }{2}} \theta \left (t - \pi \right )}{5} - \frac {9}{10}\right ) \sin {\left (t \right )}\right ) e^{- \frac {t}{2}} - \frac {4 \theta \left (t - \pi \right )}{5} + \frac {4}{5} \]