73.20.10 problem 29.7 (e)

Internal problem ID [15540]
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 (e)
Date solved : Thursday, March 13, 2025 at 06:11:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&={\mathrm e}^{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: 8.565 (sec). Leaf size: 20

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

\[ y = \frac {{\mathrm e}^{t}}{4}+\frac {\left (2 t -1\right ) {\mathrm e}^{3 t}}{4} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 68

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

\[ y(t)\to -e^{3 t} \int _1^0-e^{-2 K[1]} K[1]dK[1]+e^{3 t} \int _1^t-e^{-2 K[1]} K[1]dK[1]+\frac {1}{2} e^t \left (e^{2 t}-1\right ) t \]

✓ Sympy. Time used: 0.251 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - exp(t) - 6*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)
 

\[ y{\left (t \right )} = \left (\left (\frac {t}{2} - \frac {1}{4}\right ) e^{2 t} + \frac {1}{4}\right ) e^{t} \]