Internal problem ID [850]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing
functions. page 268
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\pi \\ 0 & \operatorname {otherwise} \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.532 (sec). Leaf size: 91
dsolve([diff(y(t),t$2)+diff(y(t),t)+5/4*y(t)=piecewise(0<=t and t<Pi,sin(t),true,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {4 \left (\left \{\begin {array}{cc} -8 \,{\mathrm e}^{-\frac {t}{4}} \left (\cos \left (t \right ) \sinh \left (\frac {t}{4}\right )-\frac {\sin \left (t \right ) \cosh \left (\frac {t}{4}\right )}{4}\right ) & t <\pi \\ \left (-{\mathrm e}^{-\frac {t}{2}+\frac {\pi }{2}}+{\mathrm e}^{-\frac {t}{2}}\right ) \left (4 \cos \left (t \right )+\sin \left (t \right )\right ) & \pi \le t \end {array}\right .\right )}{17} \]
✓ Solution by Mathematica
Time used: 0.129 (sec). Leaf size: 77
DSolve[{y''[t]+y'[t]+5/4*y[t]==Piecewise[{{Sin[t],0<=t<Pi},{0,True}}],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ \frac {4}{17} \left (\left (-4+4 e^{-t/2}\right ) \cos (t)+\left (1+e^{-t/2}\right ) \sin (t)\right ) & 0<t\leq \pi \\ -\frac {4}{17} e^{-t/2} \left (-1+e^{\pi /2}\right ) (4 \cos (t)+\sin (t)) & \text {True} \\ \end {array} \\ \end {array} \]