14.17.1 problem 19

Internal problem ID [2679]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.10, Some useful properties of Laplace transform. Excercises page 238
Problem number : 19
Date solved : Monday, January 27, 2025 at 06:06:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.780 (sec). Leaf size: 16

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

\[ y = \frac {5 \sin \left (t \right )}{2}-\frac {\cos \left (t \right ) \left (t -2\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 21

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

\[ y(t)\to \frac {5 \sin (t)}{2}-\frac {1}{2} t \cos (t)+\cos (t) \]