Internal problem ID [14566]
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: 60.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+9 \pi ^{2} y=\left \{\begin {array}{cc} 2 t & 0\le t <\pi \\ 2 t -2 \pi & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 2.875 (sec). Leaf size: 121
dsolve([diff(y(t),t$2)+9*Pi^2*y(t)=piecewise(0<=t and t<Pi,2*t,t>=Pi and t<2*Pi,2*(t-Pi),t>=2*Pi,0 ),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {2 \left (\left \{\begin {array}{cc} 0 & t <0 \\ 3 \pi t -\sin \left (3 \pi t \right ) & t <\pi \\ 3 \cos \left (3 \pi ^{2}-3 \pi t \right ) \pi ^{2}-3 \pi ^{2}+3 \pi t -\sin \left (3 \pi t \right ) & t <2 \pi \\ 3 \left (\cos \left (3 \pi ^{2}-3 \pi t \right )+\cos \left (6 \pi ^{2}-3 \pi t \right )\right ) \pi ^{2}-\sin \left (3 \pi t \right )-\sin \left (6 \pi ^{2}-3 \pi t \right ) & 2 \pi \le t \end {array}\right .\right )}{27 \pi ^{3}} \]
✓ Solution by Mathematica
Time used: 0.076 (sec). Leaf size: 146
DSolve[{y''[t]+9*Pi^2*y[t]==Piecewise[{ {2*t,0<=t<Pi},{2*(t-Pi),Pi<=t<2*Pi},{0,t>=2*Pi} }],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {2 (3 \pi t-\sin (3 \pi t))}{27 \pi ^3} & 0<t\leq \pi \\ \frac {2 \left (3 \pi (t-\pi )+3 \pi ^2 \cos (3 \pi (\pi -t))-\sin (3 \pi t)\right )}{27 \pi ^3} & \pi <t\leq 2 \pi \\ \frac {4 \cos \left (\frac {3 \pi ^2}{2}\right ) \left (3 \pi ^2 \cos \left (\frac {3}{2} \pi (3 \pi -2 t)\right )+\sin \left (\frac {3}{2} \pi (\pi -2 t)\right )-\sin \left (\frac {3}{2} \pi (3 \pi -2 t)\right )\right )}{27 \pi ^3} & t>2 \pi \\ \end {array} \\ \end {array} \]