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Differential Equations
03-25-2017, 03:29 AM
Post: #2
RE: Differential Equations
Differential equations deal with system (or formulas describing such) that undergo continuous change. In general, rather than knowing the equation a system follows, we know an equation that tells us how the equation changes. (There's a lot to this. A first course in differential equations follows a couple of courses in calculus.) If one knows what a derivative and an integral are; knows that these are inverses of each other; then the following may make sense (if I write it well enough.)

Lets take a simple example that doesn't even need calculus. We wish to describe how a car moves at constant speed. From physics, we know that the change in distance of the car, per unit time, is its velocity. So a simple equation would be S(t2-t1)=V*(t2-t1) which describes the movement of a care moving at velocity V over time t1 to t2. The distance (S for space) is the velocity times the time. Equivalently the velocity is the distance covered divided by the time if the velocity is constant. (If the font allows) this can be written as ΔS=V*Δt.

We are interested in the case where Δt is instantaneous; it's written dt and the instantaneous change in distance is written dS. Formally (without lots of necessary but tedious proof steps) we write dS=V(t)dt which means the instantaneous change in distance is the velocity at a given time an infinitesimal amount of time. (Advanced calculus courses teach the proofs and concepts necessary to get all this to work correctly.)

Now consider how to describe a car driven by a reasonably intelligent kangaroo which does not move at constant speed. Assume we are given a function describing the velocity at a given time V(t) and want the distance covered. We can solve the above differential equation dS=V(t)dt by integrating both sides with respect to distance and time respectively. The trick is to treat the differentials (carefully) as objects themselves. This gives us the equation S = V*t + C. The distance is the velocity times the elapsed time plus an undetermined constant (one always get such a constant which can be determined by boundary conditions) which in this case is the starting point.

A more interesting case (which cannot easily be solved with high-school algebra) is the formula for radioactive decay or for loss of medicine given to a person or growth of a fungus in a petri dish. In this case, physics (ok chemistry or biology) tell us that the amount of change is proportional to the amount present. Uranium loses a constant percentage each time unit; similarly for the other examples. Here the differential equation describing things is given by dX/dt = a*X with dX/dt describing the ratio of change and "a" is the constant of proportionality. By gathering Xs and ts on separate sides of the = sign we get: dX/X = a*dt. Both sides can be integrated to give Log(X) = a*t +C. Or one can exponentiate both sides and get X(t)=C*Exp(at). The constant C is the amount originally present and the decay rate is give by a. Thus proportional decay leads to an exponential loss.

Some differential equations have algebraic solutions but many have to be solved numerically. There is a bunch of lore on this too.

A couple of further examples of a more complicated nature: if one drops a brick or bomb or lawn dart from some height; the law of gravity states that the body has a constant acceleration thus dV/dt=a. Also (as above) one gets that dS/dt = V which can be combined if desired into a second order differential equation which I don't know this board's fonts well enough to show. Anyway, either way, one solves things the same way; integrate dV=adt giving V=at+C1 then pluggin this into the next equation dS/dt=V=at+C1 or dS/dt=at+C1 and gathering terms giveng dS=atdt+C1dt which can be integrated to yield S=(a*t^2)/2+C1*t+C2. C1 can be interpreted as the initial velocity and C2 as the initial position.
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Messages In This Thread
Differential Equations - bshoring - 03-25-2017, 02:13 AM
RE: Differential Equations - ttw - 03-25-2017 03:29 AM
RE: Differential Equations - Han - 03-25-2017, 04:45 AM
RE: Differential Equations - peacecalc - 03-25-2017, 02:01 PM
RE: Differential Equations - TomC - 03-26-2017, 04:36 PM
RE: Differential Equations - bshoring - 03-27-2017, 03:49 AM
RE: Differential Equations - Dieter - 03-27-2017, 08:22 AM
RE: Differential Equations - Dieter - 03-27-2017, 10:08 AM
RE: Differential Equations - Dieter - 03-27-2017, 08:57 AM
RE: Differential Equations - Han - 03-27-2017, 06:46 AM
RE: Differential Equations - Ángel Martin - 03-27-2017, 08:47 AM
RE: Differential Equations - Dieter - 03-27-2017, 09:58 AM
RE: Differential Equations - ttw - 03-27-2017, 07:27 PM
RE: Differential Equations - Han - 03-27-2017, 08:55 PM
RE: Differential Equations - Thomas Okken - 03-27-2017, 11:46 PM
RE: Differential Equations - bshoring - 03-27-2017, 11:14 PM



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