52.7.6 problem 14

Internal problem ID [8361]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.4.1 DERIVATIVES OF A TRANSFORM. Page 309
Problem number : 14
Date solved : Wednesday, March 05, 2025 at 05:35:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 1 & 0\le t <\frac {\pi }{2} \\ \sin \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.863 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+y(t) = piecewise(0 <= t and t < 1/2*Pi,1,1/2*Pi <= t,sin(t)); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left \{\begin {array}{cc} 1 & t <\frac {\pi }{2} \\ \frac {\left (-2 t +\pi \right ) \cos \left (t \right )}{4}+\sin \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right . \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 38

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (t) & t\leq 0 \\ 1 & t>0\land 2 t\leq \pi \\ \frac {1}{4} (\pi -2 t) \cos (t)+\sin (t) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.444 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((1, (t >= 0) & (t < pi/2)), (sin(t), t >= pi/2)) + y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \begin {cases} 1 & \text {for}\: t \geq 0 \wedge t < \frac {\pi }{2} \\- \frac {t \cos {\left (t \right )}}{2} + \frac {\sin {\left (t \right )}}{4} & \text {for}\: t \geq \frac {\pi }{2} \\\text {NaN} & \text {otherwise} \end {cases} \]