Internal problem ID [4949]
Book: ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG,
EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section: Chapter 6. Laplace Transforms. Problem set 6.3, page 224
Problem number: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 y^{\prime }+5 y-\left (\left \{\begin {array}{cc} 10 \sin \relax (t ) & 0<t <2 \pi \\ 0 & 2 \pi <t \end {array}\right .\right )=0} \end {gather*} With initial conditions \begin {align*} [y \left (\pi \right ) = 1, y^{\prime }\left (\pi \right ) = 2 \,{\mathrm e}^{-\pi }-2] \end {align*}
✓ Solution by Maple
Time used: 0.575 (sec). Leaf size: 86
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+5*y(t)=piecewise(0<t and t<2*Pi,10*sin(t),t>2*Pi,0),y(Pi) = 1, D(y)(Pi) = 2*exp(-Pi)-2],y(t), singsol=all)
\[ y \relax (t ) = \left \{\begin {array}{cc} \frac {{\mathrm e}^{-t} \left (3 \sin \left (2 t \right )-2 \cos \left (2 t \right )\right )}{2} & t <0 \\ \sin \left (2 t \right ) {\mathrm e}^{-t}+2 \sin \relax (t )-\cos \relax (t ) & t <2 \pi \\ \frac {\left (\sin \left (2 t \right )-2 \cos \left (2 t \right )\right ) {\mathrm e}^{2 \pi -t}}{2}+\sin \left (2 t \right ) {\mathrm e}^{-t} & 2 \pi \le t \end {array}\right . \]
✓ Solution by Mathematica
Time used: 0.097 (sec). Leaf size: 94
DSolve[{y''[t]+2*y'[t]+5*y[t]==Piecewise[{{10*Sin[t],0<t<2*Pi},{0,t>2*Pi}}],{y[Pi]==1,y'[Pi]==2*Exp[-Pi]-2}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \frac {1}{2} e^{-t} (3 \sin (2 t)-2 \cos (2 t)) & t\leq 0 \\ -\cos (t)+2 \sin (t)+e^{-t} \sin (2 t) & 0<t\leq 2 \pi \\ \frac {1}{2} e^{-t} \left (\left (2+e^{2 \pi }\right ) \sin (2 t)-2 e^{2 \pi } \cos (2 t)\right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}