76.19.5 problem 5

Internal problem ID [17729]
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 : 5
Date solved : Tuesday, January 28, 2025 at 11:01:55 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+4 y&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.706 (sec). Leaf size: 27

dsolve([diff(y(t),t$2)-2*diff(y(t),t)+4*y(t)=0,y(0) = 2, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 37

DSolve[{D[y[t],{t,2}]-2*D[y[t],t]+4*y[t]==0,{y[0]==2,Derivative[1][y][0] == 0}},y[t],t,IncludeSingularSolutions -> True]
 

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