76.18.1 problem 12

Internal problem ID [17712]
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 : 12
Date solved : Tuesday, January 28, 2025 at 11:01:43 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }-2 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 11.058 (sec). Leaf size: 28

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

\[ y = {\mathrm e}^{-t} \left (\sqrt {3}\, \sinh \left (\sqrt {3}\, t \right )+2 \cosh \left (\sqrt {3}\, t \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 48

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

\[ y(t)\to \frac {1}{2} e^{-\left (\left (1+\sqrt {3}\right ) t\right )} \left (\left (2+\sqrt {3}\right ) e^{2 \sqrt {3} t}+2-\sqrt {3}\right ) \]