46.8.6 problem 8

Internal problem ID [7377]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.4, page 230
Problem number : 8
Date solved : Sunday, March 30, 2025 at 11:55:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=10 \sin \left (t \right )+10 \delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.256 (sec). Leaf size: 47

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = 10*sin(t)+10*Dirac(t-1); 
ic:=y(0) = 1, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -10 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{2-2 t}+10 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t}-3 \cos \left (t \right )+\sin \left (t \right )-2 \,{\mathrm e}^{-2 t}+6 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.158 (sec). Leaf size: 46

ode=D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==10*(Sin[t]+DiracDelta[t-1]); 
ic={y[0]==1,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to 10 e^{1-2 t} \left (e^t-e\right ) \theta (t-1)-2 e^{-2 t}+6 e^{-t}+\sin (t)-3 \cos (t) \]

✓ Sympy. Time used: 2.342 (sec). Leaf size: 90

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-10*Dirac(t - 1) + 2*y(t) - 10*sin(t) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- 10 \int \left (\operatorname {Dirac}{\left (t - 1 \right )} + \sin {\left (t \right )}\right ) e^{2 t}\, dt + 10 \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t}\, dt + 10 \int \limits ^{0} e^{2 t} \sin {\left (t \right )}\, dt\right ) e^{- t} + 10 \int \left (\operatorname {Dirac}{\left (t - 1 \right )} + \sin {\left (t \right )}\right ) e^{t}\, dt - 10 \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{t}\, dt - 10 \int \limits ^{0} e^{t} \sin {\left (t \right )}\, dt + 1\right ) e^{- t} \]