74.11.47 problem 59

Internal problem ID [16297]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 59
Date solved : Tuesday, January 28, 2025 at 09:00:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\left \{\begin {array}{cc} 2 t & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 1.373 (sec). Leaf size: 41

dsolve([diff(y(t),t$2)+9*y(t)=piecewise(0<=t and t<Pi,2*t,t>=Pi,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {2 \left (\left \{\begin {array}{cc} 0 & t <0 \\ 3 t -\sin \left (3 t \right ) & t <\pi \\ -3 \cos \left (3 t \right ) \pi -2 \sin \left (3 t \right ) & \pi \le t \end {array}\right .\right )}{27} \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 49

DSolve[{D[y[t],{t,2}]+9*y[t]==Piecewise[{ {2*t,0<=t<Pi},{0,t>=Pi} }],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ -\frac {2}{27} (\sin (3 t)-3 t) & 0<t\leq \pi \\ -\frac {2}{27} (3 \pi \cos (3 t)+2 \sin (3 t)) & \text {True} \\ \end {array} \\ \end {array} \]