46.8.7 problem 9

Internal problem ID [7378]
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 : 9
Date solved : Wednesday, March 05, 2025 at 04:24:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=\left (1-\operatorname {Heaviside}\left (t -10\right )\right ) {\mathrm e}^{t}-{\mathrm e}^{10} \delta \left (t -10\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.547 (sec). Leaf size: 59

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = (1-Heaviside(t-10))*exp(t)-exp(10)*Dirac(t-10); 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-2 t} \left (\left (-{\mathrm e}^{3 t}+\left (\left (\sin \left (10\right )-7 \cos \left (10\right )\right ) \sin \left (t \right )+\left (7 \sin \left (10\right )+\cos \left (10\right )\right ) \cos \left (t \right )\right ) {\mathrm e}^{30}\right ) \operatorname {Heaviside}\left (t -10\right )-\cos \left (t \right )+7 \sin \left (t \right )+{\mathrm e}^{3 t}\right )}{10} \]

✓ Mathematica. Time used: 0.532 (sec). Leaf size: 94

ode=D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==(1-UnitStep[t-10])*Exp[t]-Exp[10]*DiracDelta[t-10]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{10} e^{-2 t} \left (10 e^{30} \theta (t-10) \sin (10-t)+\theta (10-t) \left (e^{3 t}+3 e^{30} \sin (10-t)-e^{30} \cos (10-t)\right )-3 e^{30} \sin (10-t)+7 \sin (t)+e^{30} \cos (10-t)-\cos (t)\right ) \]

✓ Sympy. Time used: 7.625 (sec). Leaf size: 224

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((Heaviside(t - 10) - 1)*exp(t) + Dirac(t - 10)*exp(10) + 5*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \frac {\left (e^{3 t} \sin {\left (t \right )} + 3 e^{3 t} \cos {\left (t \right )} - e^{30} \sin {\left (10 \right )} - 3 e^{30} \cos {\left (10 \right )}\right ) \theta \left (t - 10\right )}{10} + \frac {e^{3 t} \sin {\left (t \right )}}{10} + \frac {3 e^{3 t} \cos {\left (t \right )}}{10} - e^{10} \int \operatorname {Dirac}{\left (t - 10 \right )} e^{2 t} \cos {\left (t \right )}\, dt + e^{10} \int \limits ^{0} \operatorname {Dirac}{\left (t - 10 \right )} e^{2 t} \cos {\left (t \right )}\, dt + \frac {7}{10}\right ) \sin {\left (t \right )} + \left (\frac {\left (3 e^{3 t} \sin {\left (t \right )} - e^{3 t} \cos {\left (t \right )} + e^{30} \cos {\left (10 \right )} - 3 e^{30} \sin {\left (10 \right )}\right ) \theta \left (t - 10\right )}{10} - \frac {3 e^{3 t} \sin {\left (t \right )}}{10} + \frac {e^{3 t} \cos {\left (t \right )}}{10} + e^{10} \int \operatorname {Dirac}{\left (t - 10 \right )} e^{2 t} \sin {\left (t \right )}\, dt - e^{10} \int \limits ^{0} \operatorname {Dirac}{\left (t - 10 \right )} e^{2 t} \sin {\left (t \right )}\, dt - \frac {1}{10}\right ) \cos {\left (t \right )}\right ) e^{- 2 t} \]