Hi
My question is elementary. I have seen that most of the solutions to DE are either exponential or sinusoidal. I do not really understand the need to express any dynamical event with DE. Why not express with the solutions directly. For example RC circuit, output voltage can be always represented as V(1-e(-t/tau)) and we can get any desired voltage with respect to time using this equation. Why should I first represent this my DE and then solve it to make generalization? Does the above equation not suffice to describe the model of the system?
A lot of things come into play in systems that are modelled with differential equations. Let's try and discuss it a little bit. Let's discuss the initial and final values and then the exponential response.
Generally, expression of the form V(1-e^(-t/tau)) is a known solution of a system with one energy reservoir, one restriction and one 'empty' vessel if the empty vessel is to be filled with energy transferred from the reservoir through the restriction. The expression is a bit different if the vessel is not empty initially.
Realize that V(1-e^(-t/tau)) is only valid if the vessel is much smaller than the reservoir else the 'V_reservoir' in the reservoir is going to decrease exponentially, significantly as the 'V_vessel' in the vessel increases exponentially. Recall that the transfer of energy stops when V_vessel = V_reservoir. This implies that if the size of the vessel is significant relative to the size of the reservoir, then when V_vessel = V_reservoir, their value would be different from the initial value of V_reservoir. Let's take a typical illustration with two cases.
CASE 1: a constant voltage source and a capacitor
If you are charging a capacitor (with V_cap_initial = 0V) with a constant voltage source (with V_source_initial) through a resistor, when V_cap_final = V_source_final, then then the charging ends. V_source_final = V_source_initial because the charges required to fill the capacitor is insignificant relative to the charges that the source has in stock.
CASE 2: two capacitors.
Let's take two capacitors of the same size for simplicity. If you are charging a capacitor (with V_cap_vessel_initial = 0V) with another capacitor of the same size (with V_cap_reservoir_initial) then V_cap_vessel_final = V_cap_reservoir_final = V_cap_reservoir_initial/2 ideally (assuming no losses). As energy is being transferred exponentially to the vessel capacitor, Energy level in the reservoir capacitor drops exponentially too.
THE EXPONENTIAL RESPONSE
Every such system has a exponential response (sine, cosine,.... can be expressed as exponential functions -- or better put, they are exponential functions). But why exponential? It's exponential because as you begin to fill the vessel with energy, that energy that's now in the vessel begins to oppose further transfer of energy into the vessel. Energy still gets transferred as long as the energy in the vessel is not equal to that in the reservoir. The opposition gets stronger and stronger with time because more and more energy gets deposited in the vessel with time and so energy transfer becomes more and more difficult with time resulting in a lower and lower transfer rate as time passes. This happens until the energy in the vessel equals that in the reservoir and the opposition is now enough to stop any further transfer. Also notice that without the restriction, the transfer would be instantaneous.
Here, let's take a system with one reservoir, two restrictions and two vessels such that the energy transfer is as shown:
reservoir -> restriction1 -> vessel1 -> restriction2 -> vessel2
Energy from the reservoir will be transferred to vessel1 exponentially. As vessel1 is being filled up, part of the energy collected in vessel1 will be transferred exponentially to vessel2. Imagine what is happening in vessel1. It gets both exponential supply of energy and loss of energy at the same time. Also imagine what is happening vessel2. It gets an exponential supply from vessel1 whose energy level is not constant but is trying to rise exponentially (tying because it is not actually rising exponentially as part of it is collected into vessel2). This is a d
2y/dx
2 behaviour. As can be envisaged, analysing this complex behaviour would be a tedious task. The system described here is the system FvM showed in Post #5. If the components in the system were to be interconnected in a different way, its behaviour would be different.
It is noteworthy that in any interconnected systems, the transfer of energy will end simultaneously in all vessels and that the more the number of components (restrictions and storage), the longer it will take to achieve this equilibrium for the same component sizes and the complex the behaviour of the system would be and the more tedious it would be to analyze it.
Differential equation is a general, compact and systematic way to model a system. Solving the equation is results in the solution (with all necessary info contained) without having to bother about how the system behaves as energy is being transferred.
It is the solution to differential equation that is useful. Even when a differential equation is described within a system, it is described in such a way that the solutions are extracted for use. If the behaviour of nown for a system, then it's okay to go ahead and use the solutions, else it is much easier to develop the differential equation than to think about the behaviour of the system directly.