40.8.10 problem 12

Internal problem ID [8667]
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 : 12
Date solved : Tuesday, September 30, 2025 at 05:40:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=25 t -100 \delta \left (t -\pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-2 \\ y^{\prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.225 (sec). Leaf size: 27

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 25*t-100*Dirac(t-Pi); 
ic:=[y(0) = -2, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -50 \sin \left (2 t \right ) \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-t +\pi }+5 t -2 \]

✓ Mathematica. Time used: 0.189 (sec). Leaf size: 163

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==25*t-100*DiracDelta[t-Pi]; 
ic={y[0]==-2,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\frac {1}{2} e^{-t} \left (2 \sin (2 t) \int _1^0\left (\frac {25}{2} e^{K[1]} \cos (2 K[1]) K[1]-50 e^{\pi } \delta (\pi -K[1])\right )dK[1]-2 \sin (2 t) \int _1^t\left (\frac {25}{2} e^{K[1]} \cos (2 K[1]) K[1]-50 e^{\pi } \delta (\pi -K[1])\right )dK[1]+2 \cos (2 t) \int _1^0-25 e^{K[2]} \cos (K[2]) K[2] \sin (K[2])dK[2]-2 \cos (2 t) \int _1^t-25 e^{K[2]} \cos (K[2]) K[2] \sin (K[2])dK[2]-3 \sin (2 t)+4 \cos (2 t)\right ) \end{align*}

✓ Sympy. Time used: 9.656 (sec). Leaf size: 139

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-25*t + 100*Dirac(t - pi) + 5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -2, Subs(Derivative(y(t), t), t, 0): 5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (\frac {25 \int \limits ^{0} t e^{t} \sin {\left (2 t \right )}\, dt}{2} - \frac {25 \int \left (t - 4 \operatorname {Dirac}{\left (t - \pi \right )}\right ) e^{t} \sin {\left (2 t \right )}\, dt}{2} + \frac {25 \int \limits ^{0} \left (- 4 \operatorname {Dirac}{\left (t - \pi \right )} e^{t} \sin {\left (2 t \right )}\right )\, dt}{2} - 2\right ) \cos {\left (2 t \right )} + \left (- \frac {25 \int \limits ^{0} t e^{t} \cos {\left (2 t \right )}\, dt}{2} + \frac {25 \int \left (t - 4 \operatorname {Dirac}{\left (t - \pi \right )}\right ) e^{t} \cos {\left (2 t \right )}\, dt}{2} - \frac {25 \int \limits ^{0} \left (- 4 \operatorname {Dirac}{\left (t - \pi \right )} e^{t} \cos {\left (2 t \right )}\right )\, dt}{2} + \frac {3}{2}\right ) \sin {\left (2 t \right )}\right ) e^{- t} \]