67.19.5 problem 28.8 (b)

Internal problem ID [16890]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 28. The inverse Laplace transform. Additional Exercises. page 509
Problem number : 28.8 (b)
Date solved : Thursday, October 02, 2025 at 01:40:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&=t^{2} {\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: 0.083 (sec). Leaf size: 13

ode:=diff(diff(y(t),t),t)-6*diff(y(t),t)+9*y(t) = exp(3*t)*t^2; 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {t^{4} {\mathrm e}^{3 t}}{12} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 17

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

\begin{align*} y(t)&\to \frac {1}{12} e^{3 t} t^4 \end{align*}

✓ Sympy. Time used: 0.181 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2*exp(3*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)
 

\[ y{\left (t \right )} = \frac {t^{4} e^{3 t}}{12} \]