40.8.5 problem 7

Internal problem ID [8662]
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 : 7
Date solved : Tuesday, September 30, 2025 at 05:40:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+24 y^{\prime }+37 y&=17 \,{\mathrm e}^{-t}+\delta \left (t -\frac {1}{2}\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: 0.274 (sec). Leaf size: 36

ode:=4*diff(diff(y(t),t),t)+24*diff(y(t),t)+37*y(t) = 17*exp(-t)+Dirac(t-1/2); 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\left ({\mathrm e}^{\frac {3}{2}} \operatorname {Heaviside}\left (t -\frac {1}{2}\right ) \sin \left (-\frac {1}{4}+\frac {t}{2}\right )+2 \,{\mathrm e}^{2 t}+8 \sin \left (\frac {t}{2}\right )\right ) {\mathrm e}^{-3 t}}{2} \]

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 179

ode=4*D[y[t],{t,2}]+24*D[y[t],t]+27*y[t]==17*Exp[-t]+DiracDelta[t-1/2]; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\frac {1}{6} e^{-9 t/2} \left (-6 \int _1^t-\frac {1}{12} e^{\frac {7 K[1]}{2}} \left (2 e^{K[1]} \delta (2 K[1]-1)+17\right )dK[1]+6 e^{3 t} \int _1^0\frac {1}{12} e^{\frac {K[2]}{2}} \left (2 e^{K[2]} \delta (2 K[2]-1)+17\right )dK[2]-6 e^{3 t} \int _1^t\frac {1}{12} e^{\frac {K[2]}{2}} \left (2 e^{K[2]} \delta (2 K[2]-1)+17\right )dK[2]+6 \int _1^0-\frac {1}{12} e^{\frac {7 K[1]}{2}} \left (2 e^{K[1]} \delta (2 K[1]-1)+17\right )dK[1]-11 e^{3 t}+5\right ) \end{align*}

✓ Sympy. Time used: 32.888 (sec). Leaf size: 143

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1/2) + 37*y(t) + 24*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)) - 17*exp(-t),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (\frac {\int \limits ^{0} 17 e^{2 t} \sin {\left (\frac {t}{2} \right )}\, dt}{2} - \frac {\int \left (\operatorname {Dirac}{\left (t - \frac {1}{2} \right )} e^{t} + 17\right ) e^{2 t} \sin {\left (\frac {t}{2} \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {1}{2} \right )} e^{3 t} \sin {\left (\frac {t}{2} \right )}\, dt}{2} + 1\right ) \cos {\left (\frac {t}{2} \right )} + \left (- \frac {\int \limits ^{0} 17 e^{2 t} \cos {\left (\frac {t}{2} \right )}\, dt}{2} + \frac {\int \left (\operatorname {Dirac}{\left (t - \frac {1}{2} \right )} e^{t} + 17\right ) e^{2 t} \cos {\left (\frac {t}{2} \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {1}{2} \right )} e^{3 t} \cos {\left (\frac {t}{2} \right )}\, dt}{2} + 8\right ) \sin {\left (\frac {t}{2} \right )}\right ) e^{- 3 t} \]