46.7.6 problem 23

Internal problem ID [7367]
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 : 23
Date solved : Sunday, March 30, 2025 at 11:55:26 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=\left \{\begin {array}{cc} 3 \sin \left (t \right )-\cos \left (t \right ) & 0<t <2 \pi \\ 3 \sin \left (2 t \right )-\cos \left (2 t \right ) & 2 \pi <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: 0.439 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+diff(y(t),t)-2*y(t) = piecewise(0 < t and t < 2*Pi,3*sin(t)-cos(t),2*Pi < t,3*sin(2*t)-cos(2*t)); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 55

ode=D[y[t],{t,2}]+D[y[t],t]-2*y[t]==Piecewise[{{3*Sin[t]-Cos[t],0<t<2*Pi},{3*Sin[2*t]-Cos[2*t],t>2*Pi}}]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {e^{-2 t}}{3}+\frac {2 e^t}{3} & t\leq 0 \\ e^t-\sin (t) & 0<t\leq 2 \pi \\ e^t-\cos (t) \sin (t) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((3*sin(t) - cos(t), (t > 0) & (t < 2*pi)), (3*sin(2*t) - cos(2*t), t > 2*pi)) - 2*y(t) + Derivative(y(t), 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)