61.4.35 problem Problem 5(f)

Internal problem ID [15360]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 5(f)
Date solved : Thursday, October 02, 2025 at 10:12:06 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-7 y^{\prime }+6 y&=\delta \left (t -1\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: 0.104 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)-7*diff(y(t),t)+6*y(t) = Dirac(t-1); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 102

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

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

✓ Sympy. Time used: 0.626 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + 6*y(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)
 

\[ y{\left (t \right )} = \left (\left (\frac {\int \operatorname {Dirac}{\left (t - 1 \right )} e^{- 6 t}\, dt}{5} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{- 6 t}\, dt}{5}\right ) e^{5 t} - \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} e^{- t}\, dt}{5} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{- t}\, dt}{5}\right ) e^{t} \]