Internal problem ID [6683]
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. 7.3.1 TRANSLATION ON THE s-AXIS.
Page 297
Problem number: 66.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y=\left \{\begin {array}{cc} 1 & 0\le t <1 \\ 0 & 1\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.266 (sec). Leaf size: 45
dsolve([diff(y(t),t$2)+4*y(t)=piecewise(0<=t and t<1,1,t>=1,0),y(0) = 0, D(y)(0) = -1],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {\sin \left (2 t \right )}{2}+\frac {\left (\left \{\begin {array}{cc} 0 & t <0 \\ 1-\cos \left (2 t \right ) & t <1 \\ \cos \left (2 t -2\right )-\cos \left (2 t \right ) & 1\le t \end {array}\right .\right )}{4} \]
✓ Solution by Mathematica
Time used: 0.039 (sec). Leaf size: 65
DSolve[{y''[t]+4*y[t]==Piecewise[{{1,0<=t<1},{0,t>=1}}],{y[0]==0,y'[0]==-1}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -\cos (t) \sin (t) & t\leq 0 \\ \frac {1}{4} (-\cos (2 t)-2 \sin (2 t)+1) & 0<t\leq 1 \\ \frac {1}{4} (\cos (2-2 t)-\cos (2 t)-2 \sin (2 t)) & \text {True} \\ \end {array} \\ \end {array} \]