76.19.9 problem 9

Internal problem ID [17733]
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 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 11:01:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 12.321 (sec). Leaf size: 24

dsolve([diff(y(t),t$2)-2*diff(y(t),t)+2*y(t)=exp(-t),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \]

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 29

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

\[ y(t)\to \frac {1}{5} \left (e^{-t}+7 e^t \sin (t)-e^t \cos (t)\right ) \]