14.18.8 problem 8

Internal problem ID [2692]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.11, Differential equations with discontinuous right-hand sides. Excercises page 243
Problem number : 8
Date solved : Monday, January 27, 2025 at 06:09:27 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} t^{2} & 0\le t <1 \\ 0 & 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: 2.531 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 49

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} t^2+2 \cos (t)-2 & 0<t\leq 1 \\ -\cos (1-t)+2 \cos (t)-2 \sin (1-t) & t>1 \\ \end {array} \\ \end {array} \]