14.17.3 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 : Monday, January 27, 2025 at 06:06:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.837 (sec). Leaf size: 11

dsolve([diff(y(t),t$2)-2*diff(y(t),t)+y(t)=t*exp(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 15

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

\[ y(t)\to \frac {e^t t^3}{6} \]