40.8.6 problem 8

Internal problem ID [8663]
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 : Tuesday, September 30, 2025 at 05:40:24 PM
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,op(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: 0.087 (sec). Leaf size: 121

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]
 

\begin{align*} y(t)&\to -e^{-2 t} \left (-\int _1^t-10 e^{2 K[1]} (\delta (K[1]-1)+\sin (K[1]))dK[1]+e^t \int _1^010 e^{K[2]} (\delta (K[2]-1)+\sin (K[2]))dK[2]-e^t \int _1^t10 e^{K[2]} (\delta (K[2]-1)+\sin (K[2]))dK[2]+\int _1^0-10 e^{2 K[1]} (\delta (K[1]-1)+\sin (K[1]))dK[1]-e^t\right ) \end{align*}

✓ Sympy. Time used: 1.327 (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)
 

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