Week 7 - Lec01.pdf - Week 7 Lecture Notes Lec01 9.1 Differential Equations In general a differential equation is an equation that contains an unknown

# Week 7 - Lec01.pdf - Week 7 Lecture Notes Lec01 9.1...

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Week 7 Lecture Notes, Lec01 9.1. Differential Equations In general, a differential equation is an equation that contains an unknown function and one or more of its derivatives. The order of a differential equation is the order of the highest derivative that occurs in the equation. Example. ? + ? + ? ′′ = ? Differential equations can be used to create mathematical models for real-life phenomena. Population Growth It is known that the population grows at a rate proportional to the size of the population. If we define population, P, as a function of time, then we would have: ?𝑃 ?𝑡 = 𝑘𝑃(𝑡) Where k is a constant, and ?𝑃 ?𝑡 is the rate of growth of the population. Assuming there is enough nutrition, there s no epidemic diseases and no predators, the population will be always increasing; ?𝑃 ?𝑡 > 0 . In addition, the population itself is a positive number; so 𝑘 > 0 . This equation, ?𝑃 ?𝑡 = 𝑘𝑃(𝑡) , asks us to find a function whose derivative is a constant multiple of itself. A possible form of such function would be 𝑃(𝑡) = 𝐶? 𝑘𝑡 . 𝑃′(𝑡) = 𝑘(𝐶? 𝑘𝑡 ) = 𝑘𝑃(𝑡)