8.17.3 problem 21

Internal problem ID [2681]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 21
Date solved : Tuesday, September 30, 2025 at 05:49:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=t \,{\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: 0.098 (sec). Leaf size: 11

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

\[ y = \frac {t^{3} {\mathrm e}^{t}}{6} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 15

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

\begin{align*} y(t)&\to \frac {e^t t^3}{6} \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 10

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

\[ y{\left (t \right )} = \frac {t^{3} e^{t}}{6} \]