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 : Sunday, March 30, 2025 at 12:47:31 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*}

✗ Maple

ode:=diff(diff(i(t),t),t)+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); 
ic:=i(0) = 8, D(i)(0) = 0; 
dsolve([ode,ic],i(t),method='laplace');
 

\[ \text {No solution found} \]

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

ode=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}}]; 
ic={i[0]==8,Derivative[1][i][0]==0}; 
DSolve[{ode,ic},i[t],t,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
t = symbols("t") 
i = Function("i") 
ode = Eq(-Piecewise((30, (t > 0) & (t < 2*pi)), (0, (t >= 2*pi) & (t <= 5*pi)), (10, (t < oo) & (t > 5*pi))) + 3*i(t) + 2*Derivative(i(t), t) + Derivative(i(t), (t, 2)),0) 
ics = {i(0): 8, Subs(Derivative(i(t), t), t, 0): 0} 
dsolve(ode,func=i(t),ics=ics)