Internal problem ID [10989]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin.
CRC Press 2015
Section: Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number: Problem 4(e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y-\left (\left \{\begin {array}{cc} 8 t & 0\le t <\frac {\pi }{2} \\ 8 \pi & \frac {\pi }{2}\le t \end {array}\right .\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 45
dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<=t and t<Pi/2,8*t,t>=Pi/2,8*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \left \{\begin {array}{cc} 0 & t <0 \\ -\sin \left (2 t \right )+2 t & t <\frac {\pi }{2} \\ \cos \left (2 t \right ) \pi -2 \sin \left (2 t \right )+2 \pi & \frac {\pi }{2}\le t \end {array}\right . \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 47
DSolve[{y''[t]+4*y[t]==Piecewise[{{8*t,0<=t<Pi/2},{8*Pi,t>=Pi/2}}],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} 0 & t\leq 0 \\ 2 t-\sin (2 t) & t>0\land 2 t\leq \pi \\ \pi (\cos (2 t)+2)-2 \sin (2 t) & \text {True} \\ \\ \\ \\ \\ \end{align*}