76.19.8 problem 8

Internal problem ID [17732]
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 : 8
Date solved : Tuesday, January 28, 2025 at 11:01:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&=\cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.743 (sec). Leaf size: 23

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

\[ y = \frac {\cos \left (t \right ) \left (1+4 \,{\mathrm e}^{t}\right )}{5}-\frac {2 \sin \left (t \right ) \left (1+{\mathrm e}^{t}\right )}{5} \]

✓ Solution by Mathematica

Time used: 0.069 (sec). Leaf size: 29

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

\[ y(t)\to \frac {1}{5} \left (\left (4 e^t+1\right ) \cos (t)-2 \left (e^t+1\right ) \sin (t)\right ) \]