76.20.10 problem 10

Internal problem ID [17687]
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 5.6 (Differential equations with Discontinuous Forcing Functions). Problems at page 342
Problem number : 10
Date solved : Monday, March 31, 2025 at 04:24:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.297 (sec). Leaf size: 55

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+5/4*y(t) = piecewise(0 <= t and t < Pi,sin(t),Pi <= t,0); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {8 \left (\left \{\begin {array}{cc} {\mathrm e}^{-\frac {t}{4}} \left (-\sin \left (t \right ) \cosh \left (\frac {t}{4}\right )+4 \cos \left (t \right ) \sinh \left (\frac {t}{4}\right )\right ) & t <\pi \\ \left (4 \cos \left (t \right )+\sin \left (t \right )\right ) \sinh \left (\frac {\pi }{4}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}} & \pi \le t \end {array}\right .\right )}{17} \]

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 77

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ \frac {4}{17} \left (\left (-4+4 e^{-t/2}\right ) \cos (t)+\left (1+e^{-t/2}\right ) \sin (t)\right ) & 0<t\leq \pi \\ -\frac {4}{17} e^{-t/2} \left (-1+e^{\pi /2}\right ) (4 \cos (t)+\sin (t)) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

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