46.7.6 problem 14

Internal problem ID [9647]
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 : Tuesday, September 30, 2025 at 06:21:50 PM
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: 0.321 (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,op(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: 10.405 (sec). Leaf size: 170

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]
 

\begin{align*} y(t)&\to -\sin (t) \int _1^0\cos (K[3]) \left ( \begin {array}{cc} \{ & \begin {array}{cc} 1 & 0\leq K[3]<\frac {\pi }{2} \\ \sin (K[3]) & 2 K[3]\geq \pi \\ \end {array} \\ \end {array} \right )dK[3]+\sin (t) \int _1^t\cos (K[3]) \left ( \begin {array}{cc} \{ & \begin {array}{cc} 1 & 0\leq K[3]<\frac {\pi }{2} \\ \sin (K[3]) & 2 K[3]\geq \pi \\ \end {array} \\ \end {array} \right )dK[3]+\cos (t) \left (\int _1^t-\left ( \begin {array}{cc} \{ & \begin {array}{cc} 1 & 0\leq K[2]<\frac {\pi }{2} \\ \sin (K[2]) & 2 K[2]\geq \pi \\ \end {array} \\ \end {array} \right ) \sin (K[2])dK[2]-\int _1^0-\left ( \begin {array}{cc} \{ & \begin {array}{cc} 1 & 0\leq K[2]<\frac {\pi }{2} \\ \sin (K[2]) & 2 K[2]\geq \pi \\ \end {array} \\ \end {array} \right ) \sin (K[2])dK[2]+1\right ) \end{align*}

✓ Sympy. Time used: 0.255 (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)
 

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