46.8.12 problem 12

Internal problem ID [9662]
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 : Tuesday, September 30, 2025 at 06:22:01 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,op(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: 0.2 (sec). Leaf size: 167

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]
 

\begin{align*} y(t)&\to -\frac {1}{5} e^t \left (-5 \int _1^t-\frac {1}{5} e^{-K[1]} \left (\delta (K[1]-4)+e^{K[1]}+\delta (K[1]-2)\right )dK[1]+5 e^{5 t} \int _1^0\frac {1}{5} e^{-6 K[2]} \left (\delta (K[2]-4)+e^{K[2]}+\delta (K[2]-2)\right )dK[2]-5 e^{5 t} \int _1^t\frac {1}{5} e^{-6 K[2]} \left (\delta (K[2]-4)+e^{K[2]}+\delta (K[2]-2)\right )dK[2]+5 \int _1^0-\frac {1}{5} e^{-K[1]} \left (\delta (K[1]-4)+e^{K[1]}+\delta (K[1]-2)\right )dK[1]+9 e^{5 t}-54\right ) \end{align*}

✗ 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