46.7.8 problem 25

Internal problem ID [7369]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.3, page 224
Problem number : 25
Date solved : Sunday, March 30, 2025 at 11:55:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} t & 0<t <1 \\ 0 & 1<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: 0.375 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 44

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} t-\sin (t) & 0<t\leq 1 \\ \cos (1-t)-\sin (1-t)-\sin (t) & t>1 \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.388 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((t, (t > 0) & (t < 1)), (0, t > 1)) + y(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)
 

\[ y{\left (t \right )} = C_{2} \cos {\left (t \right )} + \begin {cases} t & \text {for}\: t > 0 \wedge t < 1 \\0 & \text {for}\: t > 1 \\\text {NaN} & \text {otherwise} \end {cases} \]