52.8.12 problem 12

Internal problem ID [8376]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number : 12
Date solved : Sunday, March 30, 2025 at 12:53:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-7 y^{\prime }+6 y&={\mathrm e}^{t}+\delta \left (t -2\right )+\delta \left (t -4\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.263 (sec). Leaf size: 64

ode:=diff(diff(y(t),t),t)-7*diff(y(t),t)+6*y(t) = exp(t)+Dirac(t-2)+Dirac(t-4); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-24+6 t} \operatorname {Heaviside}\left (t -4\right )}{5}+\frac {{\mathrm e}^{-12+6 t} \operatorname {Heaviside}\left (t -2\right )}{5}-\frac {{\mathrm e}^{t -4} \operatorname {Heaviside}\left (t -4\right )}{5}-\frac {{\mathrm e}^{t -2} \operatorname {Heaviside}\left (t -2\right )}{5}+\frac {{\mathrm e}^{6 t}}{25}+\frac {\left (-5 t -1\right ) {\mathrm e}^{t}}{25} \]

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 67

ode=D[y[t],{t,2}]-7*D[y[t],t]+6*y[t]==Exp[t]+DiracDelta[t-2]+DiracDelta[t-4]; 
ic={y[0]==9,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{25} e^{t-24} \left (5 \left (e^{5 t}-e^{20}\right ) \theta (t-4)+5 \left (e^{5 t+12}-e^{22}\right ) \theta (t-2)+e^{24} \left (-5 t-44 e^{5 t}+269\right )\right ) \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 4) - Dirac(t - 2) + 6*y(t) - exp(t) - 7*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out