Internal problem ID [6513]
Internal file name [OUTPUT/5761_Sunday_June_05_2022_03_53_07_PM_95044538/index.tex]
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.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_laplace", "second_order_linear_constant_coeff"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
\[
\boxed {i^{\prime \prime }+2 i^{\prime }+3 i=\left \{\begin {array}{cc} 30 & 0
This is a linear ODE. In canonical form it is written as \begin {align*} i^{\prime \prime } + p(t)i^{\prime } + q(t) i &= F \end {align*}
Where here \begin {align*} p(t) &=2\\ q(t) &=3\\ F &=\left \{\begin {array}{cc} 0 & t \le 0 \\ 30 & t <2 \pi \\ 0 & t \le 5 \pi \\ 10 & 5 \pi Hence the ode is \begin {align*} i^{\prime \prime }+2 i^{\prime }+3 i = \left \{\begin {array}{cc} 0 & t \le 0 \\ 30 & t <2 \pi \\ 0 & t \le 5 \pi \\ 10 & 5 \pi The domain of \(p(t)=2\) is \[
\{-\infty Solving using the Laplace transform method. Let \begin {align*} \mathcal {L}\left (i\right ) =Y(s) \end {align*}
Taking the Laplace transform of the ode and using the relations that \begin {align*} \mathcal {L}\left (i^{\prime }\right ) &= s Y(s) - i \left (0\right )\\ \mathcal {L}\left (i^{\prime \prime }\right ) &= s^2 Y(s) - i'(0) - s i \left (0\right ) \end {align*}
The given ode now becomes an algebraic equation in the Laplace domain \begin {align*} s^{2} Y \left (s \right )-i^{\prime }\left (0\right )-s i \left (0\right )+2 s Y \left (s \right )-2 i \left (0\right )+3 Y \left (s \right ) = \frac {30-30 \,{\mathrm e}^{-2 \pi s}+10 \,{\mathrm e}^{-5 \pi s}}{s}\tag {1} \end {align*}
But the initial conditions are \begin {align*} i \left (0\right )&=8\\ i'(0) &=0 \end {align*}
Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )-16-8 s +2 s Y \left (s \right )+3 Y \left (s \right ) = \frac {30-30 \,{\mathrm e}^{-2 \pi s}+10 \,{\mathrm e}^{-5 \pi s}}{s} \end {align*}
Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = -\frac {2 \left (-4 s^{2}+15 \,{\mathrm e}^{-2 \pi s}-5 \,{\mathrm e}^{-5 \pi s}-8 s -15\right )}{s \left (s^{2}+2 s +3\right )} \end {align*}
Taking the inverse Laplace transform gives \begin {align*} i&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (-\frac {2 \left (-4 s^{2}+15 \,{\mathrm e}^{-2 \pi s}-5 \,{\mathrm e}^{-5 \pi s}-8 s -15\right )}{s \left (s^{2}+2 s +3\right )}\right )\\ &= 10-{\mathrm e}^{-t} \left (\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )+2 \cos \left (\sqrt {2}\, t \right )\right )+\frac {5 \left (i \sqrt {2}+2\right ) \left (4-2 i \sqrt {2}-3 \,{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -5 \pi \right )}-\left (1-2 i \sqrt {2}\right ) {\mathrm e}^{-\left (-i \sqrt {2}+1\right ) \left (t -5 \pi \right )}\right ) \operatorname {Heaviside}\left (t -5 \pi \right )}{18}+\frac {5 \left (i \sqrt {2}+2\right ) \left (-4+2 i \sqrt {2}+3 \,{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}+\left (1-2 i \sqrt {2}\right ) {\mathrm e}^{-\left (-i \sqrt {2}+1\right ) \left (t -2 \pi \right )}\right ) \operatorname {Heaviside}\left (t -2 \pi \right )}{6} \end {align*}
Converting the above solution to piecewise it becomes \[
i = \left \{\begin {array}{cc} 10-{\mathrm e}^{-t} \left (\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )+2 \cos \left (\sqrt {2}\, t \right )\right ) & t <2 \pi \\ 10-{\mathrm e}^{-t} \left (\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )+2 \cos \left (\sqrt {2}\, t \right )\right )+\frac {5 \left (i \sqrt {2}+2\right ) \left (-4+2 i \sqrt {2}+3 \,{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}+\left (1-2 i \sqrt {2}\right ) {\mathrm e}^{-\left (-i \sqrt {2}+1\right ) \left (t -2 \pi \right )}\right )}{6} & t <5 \pi \\ 10-{\mathrm e}^{-t} \left (\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )+2 \cos \left (\sqrt {2}\, t \right )\right )+\frac {5 \left (i \sqrt {2}+2\right ) \left (4-2 i \sqrt {2}-3 \,{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -5 \pi \right )}-\left (1-2 i \sqrt {2}\right ) {\mathrm e}^{-\left (-i \sqrt {2}+1\right ) \left (t -5 \pi \right )}\right )}{18}+\frac {5 \left (i \sqrt {2}+2\right ) \left (-4+2 i \sqrt {2}+3 \,{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}+\left (1-2 i \sqrt {2}\right ) {\mathrm e}^{-\left (-i \sqrt {2}+1\right ) \left (t -2 \pi \right )}\right )}{6} & 5 \pi \le t \end {array}\right .
\] Simplifying the solution gives \[
i = -\sqrt {2}\, {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )-2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )+5 \left (\left \{\begin {array}{cc} 2 & t <2 \pi \\ -\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}}{2}+\frac {i \sqrt {2}\, {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}}{2}+{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}+{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )} & t <5 \pi \\ \frac {\left (-i \sqrt {2}-2\right ) {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -5 \pi \right )}}{6}-\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}}{2}+\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -5 \pi \right )}}{6}+\frac {i \sqrt {2}\, {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}}{2}+{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}-\frac {{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -5 \pi \right )}}{3}+{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}+\frac {2}{3} & 5 \pi \le t \end {array}\right .\right )
\]
The solution(s) found are the following \begin{align*}
\tag{1} i &= -\sqrt {2}\, {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )-2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )+5 \left (\left \{\begin {array}{cc} 2 & t <2 \pi \\ -\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}}{2}+\frac {i \sqrt {2}\, {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}}{2}+{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}+{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )} & t <5 \pi \\ \frac {\left (-i \sqrt {2}-2\right ) {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -5 \pi \right )}}{6}-\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}}{2}+\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -5 \pi \right )}}{6}+\frac {i \sqrt {2}\, {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}}{2}+{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}-\frac {{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -5 \pi \right )}}{3}+{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}+\frac {2}{3} & 5 \pi \le t \end {array}\right .\right ) \\
\end{align*} Verification of solutions
\[
i = -\sqrt {2}\, {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )-2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )+5 \left (\left \{\begin {array}{cc} 2 & t <2 \pi \\ -\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}}{2}+\frac {i \sqrt {2}\, {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}}{2}+{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}+{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )} & t <5 \pi \\ \frac {\left (-i \sqrt {2}-2\right ) {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -5 \pi \right )}}{6}-\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}}{2}+\frac {i \sqrt {2}\, {\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -5 \pi \right )}}{6}+\frac {i \sqrt {2}\, {\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}}{2}+{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -2 \pi \right )}-\frac {{\mathrm e}^{\left (-1+i \sqrt {2}\right ) \left (t -5 \pi \right )}}{3}+{\mathrm e}^{-\left (i \sqrt {2}+1\right ) \left (t -2 \pi \right )}+\frac {2}{3} & 5 \pi \le t \end {array}\right .\right )
\] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [i^{\prime \prime }+2 i^{\prime }+3 i=\left \{\begin {array}{cc} 0 & t \le 0 \\ 30 & t <2 \pi \\ 0 & t \le 5 \pi \\ 10 & 5 \pi Maple trace
✗ Solution by Maple
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.234 (sec). Leaf size: 297
\[
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 ) & 026.1.1 Existence and uniqueness analysis
26.1.2 Maple step by step solution
`Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
trying high order exact linear fully integrable
trying differential order: 2; linear nonhomogeneous with symmetry [0,1]
trying a double symmetry of the form [xi=0, eta=F(x)]
-> Try solving first the homogeneous part of the ODE
checking if the LODE has constant coefficients
<- constant coefficients successful
<- solving first the homogeneous part of the ODE successful`
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)
DSolve[{i''[t]+2*i'[t]+3*i[t]==Piecewise[{{30,0<t<2*Pi},{0,2*Pi<= t <= 5*Pi},{10,5*Pi<t<Infinity}}],{i[0]==8,i'[0]==0}},i[t],t,IncludeSingularSolutions -> True]