76.18.5 problem 16

Internal problem ID [17716]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 11:01:45 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&=t^{2} {\mathrm e}^{t}+7 \end{align*}

Using Laplace method With initial conditions

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

Solution by Maple

Time used: 11.988 (sec). Leaf size: 25

dsolve([diff(y(t),t$2)-2*diff(y(t),t)+2*y(t)=t^2*exp(t)+7,y(0) = 1, D(y)(0) = 1],y(t), singsol=all)
 
\[ y = \frac {7}{2}+\frac {\left (2 t^{2}-\cos \left (t \right )+7 \sin \left (t \right )-4\right ) {\mathrm e}^{t}}{2} \]

Solution by Mathematica

Time used: 0.360 (sec). Leaf size: 38

DSolve[{D[y[t],{t,2}]-2*D[y[t],t]+2*y[t]==t^2*Exp[t]+7,{y[0]==1,Derivative[1][y][0] == 1}},y[t],t,IncludeSingularSolutions -> True]
 
\[ y(t)\to \frac {1}{2} \left (2 e^t t^2-4 e^t+7 e^t \sin (t)-e^t \cos (t)+7\right ) \]