76.18.13 problem 25

Internal problem ID [17724]
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 : 25
Date solved : Tuesday, January 28, 2025 at 11:01:52 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} t & 0\le t <1 \\ 1 & 1\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 16.464 (sec). Leaf size: 31

dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<=t and t<1,t,t>=1,1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\sin \left (2 t \right )}{8}+\frac {\left (\left \{\begin {array}{cc} t & t <1 \\ 1+\frac {\sin \left (2 t -2\right )}{2} & 1\le t \end {array}\right .\right )}{4} \]

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 39

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ t-\sin (t) & 0<t\leq 1 \\ -\sin (1-t)-\sin (t)+1 & \text {True} \\ \end {array} \\ \end {array} \]