46.8.8 problem 8

Internal problem ID [9658]
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 : 8
Date solved : Tuesday, September 30, 2025 at 06:21:57 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }&=1+\delta \left (t -2\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: 0.142 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 60.045 (sec). Leaf size: 146

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

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

✓ Sympy. Time used: 0.561 (sec). Leaf size: 73

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

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