73.20.9 problem 29.7 (d)

Internal problem ID [15539]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 29. Convolution. Additional Exercises. page 523
Problem number : 29.7 (d)
Date solved : Thursday, March 13, 2025 at 06:11:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&={\mathrm e}^{-3 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: 8.810 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)-6*diff(y(t),t)+9*y(t) = exp(-3*t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {t \cosh \left (3 t \right )}{6}+\frac {\sinh \left (3 t \right ) \left (-1+3 t \right )}{18} \]

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 70

ode=D[y[t],{t,2}]-6*D[y[t],t]+9*y[t]==Exp[-3*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -e^{3 t} \int _1^0-e^{-6 K[1]} K[1]dK[1]+e^{3 t} \int _1^t-e^{-6 K[1]} K[1]dK[1]+\frac {1}{6} e^{-3 t} \left (e^{6 t}-1\right ) t \]

✓ Sympy. Time used: 0.275 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-3*t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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