78.26.2 problem 4 (b)

Internal problem ID [18384]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 9. Laplace transforms. Section 53. More about Convolutions. The Unit Step and Impulse Functions. Problems at page 481
Problem number : 4 (b)
Date solved : Thursday, March 13, 2025 at 11:55:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-6 y&=t \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: 0.158 (sec). Leaf size: 31

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

\[ y = -\frac {\left (-9 \,{\mathrm e}^{5 t}+30 t \,{\mathrm e}^{3 t}+5 \,{\mathrm e}^{3 t}+4\right ) {\mathrm e}^{-3 t}}{180} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 28

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

\[ y(t)\to \frac {1}{180} \left (-30 t-4 e^{-3 t}+9 e^{2 t}-5\right ) \]

✓ Sympy. Time used: 0.180 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = - \frac {t}{6} + \frac {e^{2 t}}{20} - \frac {1}{36} - \frac {e^{- 3 t}}{45} \]