46.7.9 problem 26

Internal problem ID [7370]
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 : 26
Date solved : Sunday, March 30, 2025 at 11:55:33 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=\left \{\begin {array}{cc} 10 \sin \left (t \right ) & 0<t <2 \pi \\ 0 & 2 \pi <t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (\pi \right )&=1\\ y^{\prime }\left (\pi \right )&=2 \,{\mathrm e}^{-\pi }-2 \end{align*}

✓ Maple. Time used: 0.396 (sec). Leaf size: 70

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = piecewise(0 < t and t < 2*Pi,10*sin(t),2*Pi < t,0); 
ic:=y(Pi) = 1, D(y)(Pi) = 2*exp(-Pi)-2; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 94

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==Piecewise[{{10*Sin[t],0<t<2*Pi},{0,t>2*Pi}}]; 
ic={y[Pi]==1,Derivative[1][y][Pi]==2*Exp[-Pi]-2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-t} (3 \sin (2 t)-2 \cos (2 t)) & t\leq 0 \\ -\cos (t)+2 \sin (t)+e^{-t} \sin (2 t) & 0<t\leq 2 \pi \\ \frac {1}{2} e^{-t} \left (\left (2+e^{2 \pi }\right ) \sin (2 t)-2 e^{2 \pi } \cos (2 t)\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((10*sin(t), (t > 0) & (t < 2*pi)), (0, t > 2*pi)) + 5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(pi): 1, Subs(Derivative(y(t), t), t, pi): -2 + 2*exp(-pi)} 
dsolve(ode,func=y(t),ics=ics)