50.26.1 problem 3

Internal problem ID [8181]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 7. Laplace Transforms. Section 7.5 Problesm for review and discovery. Section B, Challenge Problems. Page 310
Problem number : 3
Date solved : Monday, January 27, 2025 at 03:45:18 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} i^{\prime \prime }+2 i^{\prime }+3 i&=\left \{\begin {array}{cc} 30 & 0<t <2 \pi \\ 0 & 2 \pi \le t \le 5 \pi \\ 10 & 5 \pi <t <\infty \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} i \left (0\right )&=8\\ i^{\prime }\left (0\right )&=0 \end{align*}

✗ Solution by Maple

dsolve([diff(i(t),t$2)+2*diff(i(t),t)+3*i(t)=piecewise(0<t and t<2*Pi,30,2*Pi<= t and t<= 5*Pi,0,5*Pi<t and t<infinity,10),i(0) = 8, D(i)(0) = 0],i(t), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.235 (sec). Leaf size: 297

DSolve[{D[i[t],{t,2}]+2*D[i[t],t]+3*i[t]==Piecewise[{{30,0<t<2*Pi},{0,2*Pi<= t <= 5*Pi},{10,5*Pi<t<Infinity}}],{i[0]==8,Derivative[1][i][0]==0}},i[t],t,IncludeSingularSolutions -> True]
 

\[ i(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} \left (-2 \cos \left (\sqrt {2} t\right )+10 e^t-\sqrt {2} \sin \left (\sqrt {2} t\right )\right ) & 0<t\leq 2 \pi \\ 4 e^{-t} \left (2 \cos \left (\sqrt {2} t\right )+\sqrt {2} \sin \left (\sqrt {2} t\right )\right ) & t\leq 0 \\ -e^{-t} \left (2 \cos \left (\sqrt {2} t\right )-10 e^{2 \pi } \cos \left (\sqrt {2} (t-2 \pi )\right )+\sqrt {2} \left (\sin \left (\sqrt {2} t\right )-5 e^{2 \pi } \sin \left (\sqrt {2} (t-2 \pi )\right )\right )\right ) & 2 \pi <t\leq 5 \pi \\ \frac {1}{3} e^{-t} \left (-6 \cos \left (\sqrt {2} t\right )+10 e^t-10 e^{5 \pi } \cos \left (\sqrt {2} (t-5 \pi )\right )+30 e^{2 \pi } \cos \left (\sqrt {2} (t-2 \pi )\right )-3 \sqrt {2} \sin \left (\sqrt {2} t\right )-5 \sqrt {2} e^{5 \pi } \sin \left (\sqrt {2} (t-5 \pi )\right )+15 \sqrt {2} e^{2 \pi } \sin \left (\sqrt {2} (t-2 \pi )\right )\right ) & \text {True} \\ \end {array} \\ \end {array} \]