67.18.9 problem 27.1 (i)

Internal problem ID [16880]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (i)
Date solved : Thursday, October 02, 2025 at 01:40:04 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+6 y&=t^{2} {\mathrm e}^{4 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 27

ode:=diff(diff(y(t),t),t)-5*diff(y(t),t)+6*y(t) = t^2*exp(4*t); 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {7 \,{\mathrm e}^{2 t}}{4}+\frac {\left (2 t^{2}-6 t +7\right ) {\mathrm e}^{4 t}}{4} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 32

ode=D[y[t],{t,2}]-5*D[y[t],t]+6*y[t]==t^2*Exp[4*t]; 
ic={y[0]==0,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{4} e^{2 t} \left (e^{2 t} \left (2 t^2-6 t+7\right )-7\right ) \end{align*}

✓ Sympy. Time used: 0.203 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2*exp(4*t) + 6*y(t) - 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {\left (2 t^{2} - 6 t + 7\right ) e^{2 t}}{4} - \frac {7}{4}\right ) e^{2 t} \]