Internal problem ID [5939]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 7 THE LAPLACE TRANSFORM. 7.4.1 DERIVATIVES OF A TRANSFORM.
Page 309
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+16 y-\left (\left \{\begin {array}{cc} \cos \left (4 t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right .\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.378 (sec). Leaf size: 25
dsolve([diff(y(t),t$2)+16*y(t)=piecewise(0<=t and t<Pi,cos(4*t),t>= Pi,0),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
\[ y \relax (t ) = \frac {\sin \left (4 t \right ) \left (2+\left (\left \{\begin {array}{cc} 0 & t <0 \\ t & t <\pi \\ \pi & \pi \le t \end {array}\right .\right )\right )}{8} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 60
DSolve[{y''[t]+16*y[t]==Piecewise[{{Cos[4*t],0<=t<Pi},{0,t>=Pi}}],{y[0]==1,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \cos (4 t)+\frac {1}{4} \sin (4 t) & t\leq 0 \\ \cos (4 t)+\frac {1}{8} (2+\pi ) \sin (4 t) & t>\pi \\ \cos (4 t)+\frac {1}{8} (t+2) \sin (4 t) & \text {True} \\ \\ \\ \\ \\ \end{align*}