@ -100,7 +100,11 @@ Differential equations are generally concerned about three questions:

\item If a differential equation does have a solution can we find it?

\end{enumerate}

There are two types of differential equations. \textit{Ordinary differential equations} (PDE) and \textit{Partial differential equations}. Our study won't go into further detail about PDE and will stay focused mainly on ODE.

The \textbf{order} or the differential equation is the highest derivative contained within it. \textbf{Degree} is the exponent on that highest derivative.

There are multiple ways to solve differential equations. From the numerical ones, notable are Euler's method and Runge-Kutta (RK4). Some other are described briefly in the following sections.

Differential equations fall to two groups - \textit{ordinary differential equations} (PDE) and \textit{partial differential equations}. Our study won't go into further detail about PDE and will stay focused mainly on ODE.

\subsection{Direction Field}

@ -158,7 +162,15 @@ If the system is perturbed slightly, it always returns to the standard cycle. Li

\subsection{Damping}

Mentioning damping is important mainly because, in a real world, oscillations eventually stop, due to Newton's law of Thermodynamics (the frictional force). In electronics, there is no ideal oscillator, too - small amount of energy is lost every cycle, due to electric resistance.

Generally, the damping is linear either linear or nonlinear. As a rule of thumb, the linear one is easily modeled mathematically, obeying known rules, while the nonlinear one is not \cite{institute1989estimation}. There are some use cases, where nonlinear damping is advantageous, but the research is still ongoing about this topic.

Generally, the damping is linear either linear or nonlinear. As a rule of thumb, the linear one is easily modeled mathematically, obeying known rules, while the nonlinear one is not \cite{institute1989estimation}. Nonlinear damping is advantageous in multiple cases and the research is still ongoing about this topic.

There are four exclusive states, that damping in a system can be in:

\begin{enumerate}

\item Undamped

\item Underdumped

\item Critically-dumped

\item Overdumped

\end{enumerate}

\section{Advanced Studies}

@ -292,10 +304,26 @@ $$i=\frac{V}{R}\,e^{-t/RC}$$

\section{Second-order Electrical Circuits}

adsfsa

Second order circuits contain both nonlinear elements. A RLC circuit consist of the resistor, the inductor and the capacitor and is shown in \cref{f:rlc}.

\begin{figure}[ht!]

\centering

\includegraphics[width=.75\linewidth]{rlc}

\caption{RLC circuit diagram.}

\label{f:rlc}

\end{figure}

In order to follow further, we must define new term, that will be used. \textbf{Electro-motive force} (EMF) is the force, that moves electrons from lower potential to the higher one, as opposed to so far mentioned electric potential, that can do it only in reverse order. The source of EMF can be for instance chemical reaction in cell battery, that induces the \textit{separation of charge}.

The EMF is mentioned, because nonlinear elements (capacitor and inductor) store and release energy, as well as cells do. Thus they have the ability to move electrons from one potential to another and so they have to be described in terms of electro-magnetic force, instead of just electric potential.

If the driving force in RLC circuit is \textbf{constant}, the current equation for a circuit is

$$L\,\frac{di}{dt}+ Ri +\frac{1}{C}\int i\,dt = E$$

which is the second order linear differential equation (homogenous).

The circuit itself is a damped oscillator. Writing the equation in its auxiliary form and finding its roots, we could obtain a formula for it's \textit{damping factor}, however, it is a topic far off the boundaries of this work.

\section{Conclusion}

The conclusion goes here.

@ -316,14 +344,14 @@ The conclusion goes here.

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\appendices

\section{Proof of the First Zonklar Equation}

Appendix one text goes here.

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% if you want by leaving the argument blank

\section{}

Appendix two text goes here.

%\appendices

%\section{Proof of the First Zonklar Equation}

%Appendix one text goes here.

%

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