Internal problem ID [842]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.2, The Laplace Transform. Solution of Initial Value Problems. page
255
Problem number: 17.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y-\left (\left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t <\infty \end {array}\right .\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 30
dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<=t and t<Pi,1,Pi<=t and t<infinity,0),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \left \{\begin {array}{cc} \cos \left (2 t \right ) & t <0 \\ \frac {1}{4}+\frac {3 \cos \left (2 t \right )}{4} & t <\pi \\ \cos \left (2 t \right ) & \pi \le t \end {array}\right . \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 31
DSolve[{y''[t]+4*y[t]==Piecewise[{{1,0<t<Pi},{0,Pi<=t<Infinity}}],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \cos (2 t) & t>\pi \lor t\leq 0 \\ \frac {1}{4} (3 \cos (2 t)+1) & \text {True} \\ \\ \\ \\ \\ \end{align*}