67.20.7 problem 29.7 (b)

Internal problem ID [16903]
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 (b)
Date solved : Thursday, October 02, 2025 at 01:40:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 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: 0.103 (sec). Leaf size: 23

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

\[ y = \frac {2 \,{\mathrm e}^{\frac {3 t}{2}} \left (t \cosh \left (\frac {3 t}{2}\right )-\frac {2 \sinh \left (\frac {3 t}{2}\right )}{3}\right )}{9} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 25

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

\begin{align*} y(t)&\to \frac {1}{27} \left (3 t+e^{3 t} (3 t-2)+2\right ) \end{align*}

✓ Sympy. Time used: 0.122 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = \frac {t}{9} + \left (\frac {t}{9} - \frac {2}{27}\right ) e^{3 t} + \frac {2}{27} \]