**Real Life Applications of Differential Equations**

Differential equations in electrical engineering

•Differential equations (DE's) are used to describe the behaviour of circuits containing energy storage components - capacitors and inductors. The order of the DE equates to the number of such storage elements in the circuit - either in series or in parallel.

The easiest example is a series RC network. One resistor and one capacitor in series with a voltage source Vs

So Vs = Vc + Vr

Vc is dependent on the time integral of the current, while Vr is directly proportional to the current. So try solving for the current, and you will end up with a simple first order DE.

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Rate Laws

differential equations used to explore the relationships between species and the total size of populations

•Differential equations can describe the relationship of predators and prey in an ecosystem, the rates of hybrid selection or the consequences of over-population, under-population or over-harvesting in various species. Medical science uses similar equations to determine rates of growth of tumors and to try to understand cancer growth.

•A population whose size increases linearly in time would have a constant population growth rate given by

Growth rate of population = (Nt -N0) / (t -t0) = dN/dt = constant

•Exponential population growth is described by the simple differential equation

dN/dt = bN - dN = (b - d)N = rN

where, again, b is the instantaneous birth rate per individual and d the instantaneous death rate per individual (remember that r = b – d).

What is a differential equation?

A differential equation is a mathematical expression involving the derivatives of variables – a strict definition

•They are used to model real life situations

•Uses?

Physics – Schrodinger, atomic physics

Chemistry – Rate Laws, Statistical Thermo

…Synthetic Biology (?)

Differential equations in Chemistry

•Most chemical reactions that are important for a scientific understanding of our world involve complex mechanisms, the development of mathematical models of reactions must be preceded by a substantial amount of theoretical and experimental work in Chemistry aimed at gathering an understanding of the mechanisms. In this module, we will restrict our attention to the study of simple chemical reactions. Simple reactions are reactions that do not involve complex mechanisms. The study of simple reactions is a good starting point for learning some of the mathematics that also pertains to the study of more complex reactions.

Differentiation equations in Economics

•Economists use differential equations to describe investment returns, sales practices, equilibrium and stability in economic markets and even the results of advertising in the Sethi advertising model.

•You are always differentiating to find ‘marginals‘. The concept of ‘marginals’ (marginal revenue, marginal product, marginal cost) etc is about the most important concept in microeconomics, because all decisions are taken ‘at the margin’. Do you increase production by another unit or just produce at the level you are doing? Well if your marginal revenue (the amount of revenue you will earn by producing another unit of output) is higher than your marginal cost (the amount it will cost you to produce another unit) then go for it. If your marginal cost is higher then you don’t. As you produce more your MR will fall and your MC will rise so you will maximise profits by producing where MR = MC. Basic golden rule of micro! Because MR is basically the ‘change in revenue over the change in output’ you find it by differentiating total revenue with respect to output. Total revenue is price x quantity.

The Rate Law

•An important concept of chemistry that is crucial to the development of mathematical models of chemical reactions is the Rate Law. For the homogeneous reaction

•aA+bB� cC+dD

•with reaction rate v (as defined in the previous section), the Rate Law gives the equation

•v( t) = k[ A] a[ B] b

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Pictures

**Done By: Hala Saeed, Hind Khalifa & Aisha Abdallah**

•We can model this by the differential equation: