Modelling with First-Order Equations: every key term you need (+ practice quiz)
25 flashcard terms for Differential Equations Topic 2, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A differential equation together with a stated correspondence between its symbols and measurable quantities, plus the assumptions that justify the balance law used. A model is only as trustworthy as the assumptions that were used to build it.
Conservation balance
The modelling principle that the rate of change of a stored quantity equals the rate in minus the rate out. Nearly every applied first-order equation in this course is a written version of that one sentence.
Exponential growth model
The equation stating that the rate of change of a population is proportional to the population itself. It predicts unbounded geometric increase and is realistic only while resources remain effectively unlimited.
Exponential decay and half-life
Negative proportional growth, with solutions falling by a constant factor over equal time steps. The half-life is the time for the amount to halve and equals the natural logarithm of two divided by the decay rate.
Radiocarbon dating
An application of exponential decay in which the measured ratio of a radioactive isotope to a stable one is inverted through the decay law to estimate elapsed time since the sample stopped exchanging with its surroundings.
Logistic growth equation
Growth proportional to the population multiplied by the unused fraction of the carrying capacity. It reproduces early near-exponential increase and later saturation, and is separable by partial fractions.
Carrying capacity
The population level at which the logistic growth rate falls to zero, so that the environment supports no further net increase. It is the nonzero equilibrium of the logistic equation and attracts all positive initial data.
Logistic inflection point
The moment when the logistic population passes half the carrying capacity, where the growth rate is largest and the solution curve changes from concave up to concave down, giving the S-shaped profile.
Newton law of cooling
The assumption that an object changes temperature at a rate proportional to the difference between its temperature and the surrounding medium. It gives a first-order linear equation whose solutions approach ambient temperature.
Ambient temperature
The temperature of the surrounding medium in a cooling problem, treated as a large reservoir unaffected by the object. It is the equilibrium the solution approaches, and it may itself vary with time in a richer model.
Mixing tank problem
A model tracking dissolved substance in a well-stirred tank with inflow and outflow. The rate in is inflow concentration times inflow rate, and the rate out is current concentration times outflow rate.
Well-stirred assumption
The idealisation that a tank contents are uniform in composition at every instant, so the outflow concentration equals the tank average. Without it the model would need spatial variation and a partial differential equation.
Unequal inflow and outflow
When the flow rates differ, the volume in the tank changes linearly with time, so the coefficient in the outflow term is a function of time rather than a constant, and the equation stays linear but not constant-coefficient.
Free fall with linear drag
A velocity model in which acceleration equals gravitational acceleration minus a constant times the velocity. It is first-order linear in velocity and its solutions rise to a horizontal asymptote.
Terminal velocity
The equilibrium speed at which drag exactly balances gravity, found by setting the acceleration to zero. Every solution of the drag model approaches it, from above or below, without ever crossing it.
Quadratic drag model
A falling-body model in which the resisting force is proportional to the square of the speed, appropriate at higher speeds. The equation is nonlinear but separable, and it still has a terminal velocity.
Compound interest equation
A model in which an account balance grows at a rate proportional to the balance, optionally with a continuous deposit or withdrawal stream. The deposit stream makes it linear with a forcing term rather than purely proportional.
Continuous annuity
A steady flow of payments modelled as a constant term added to a proportional-growth equation. The integrating factor solution splits the balance into a growing initial part and an accumulating contributions part.
Escape velocity model
A first-order equation obtained by treating velocity as a function of distance and using the inverse-square gravitational field. Requiring the speed to remain positive at arbitrarily large distance yields the escape criterion.
Dimensional consistency check
Verifying that each term of a modelled equation carries the same physical units. A mismatch reveals a missing rate constant or a misplaced factor before any solving effort is wasted.
Rate constant units
The units a proportionality constant must carry so that its product with the modelled quantity has units of a rate. In proportional growth it is inverse time, which is why its reciprocal sets the natural timescale.
Nondimensionalisation
Rescaling the dependent and independent variables by characteristic values so the equation contains as few parameters as possible. It reveals which parameter combinations actually control behaviour.
Steady state of a model
A solution that stays constant because the modelled inflow and outflow balance exactly. Long-time behaviour of many linear models is a steady state plus a transient that decays away.
Transient term
The part of a modelled solution that carries the memory of the initial condition and decays toward zero. Once it is negligible the system is described by the steady state alone, which is why the transient sets the settling time.
Model validation
Comparing predictions against data not used to build or fit the model, and revising assumptions when they disagree. Good agreement over a fitted range is weak evidence compared with successful prediction outside it.