61.4.4 problem Problem 2(d)

Internal problem ID [15329]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(d)
Date solved : Thursday, October 02, 2025 at 10:11:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.090 (sec). Leaf size: 15

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+4*y(t) = t^2*exp(2*t); 
ic:=[y(0) = 1, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{2 t} \left (t^{4}+12\right )}{12} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 19

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

\begin{align*} y(t)&\to \frac {1}{12} e^{2 t} \left (t^4+12\right ) \end{align*}

✓ Sympy. Time used: 0.186 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2*exp(2*t) + 4*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {t^{4}}{12} + 1\right ) e^{2 t} \]