Internal problem ID [843]
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: 18.
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 <1 \\ 0 & 1\le t <\infty \end {array}\right .\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.5 (sec). Leaf size: 38
dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<=t and t<1,1,1<=t and t<infinity,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \frac {\left (\left \{\begin {array}{cc} 0 & t <0 \\ 1-\cos \left (2 t \right ) & t <1 \\ \cos \left (2 t -2\right )-\cos \left (2 t \right ) & 1\le t \end {array}\right .\right )}{4} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 39
DSolve[{y''[t]+4*y[t]==Piecewise[{{1,0<t<1},{0,1<=t<Infinity}}],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} 0 & t\leq 0 \\ \frac {\sin ^2(t)}{2} & 0<t\leq 1 \\ -\frac {1}{2} \sin (1) \sin (1-2 t) & \text {True} \\ \\ \\ \\ \\ \end{align*}