Modelling with Differential Equations

Name: Modelling with Differential Equations
Author: StemPy Lab

How long does a cup of tea take to cool? How does the salt in a tank change while water flows in and out? Turn each story into a differential equation, solve it and read the answer back in words. You model growth and decay, cooling and mixing, and curves that cross every curve of a family at right angles. Python follows a solution step by step with Euler's method. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 40 min5 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 35 min
Growth and decay
When the rate of change of a quantity is proportional to the amount present, the equation is $\dfrac{dN}{dt} = kN$ and the solution is $N = N_0 e^{kt}$ . A positive $k$ gives growth, with a doubling time $\dfrac{\ln 2}{k}$ . A negative $k$ gives decay, with a half-life. The logistic equation adds a limit, and Euler's method follows any such equation step by step.
2 simulations1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 35 min
Cooling and mixing
Newton's law of cooling says that an object loses heat at a rate proportional to the gap between its temperature and the temperature of its surroundings: $\dfrac{dT}{dt} = -k(T - T_s)$ . A mixing problem balances the salt, or any dissolved substance, flowing in against the salt flowing out. Both give a first-order equation whose solution settles to a steady state.
2 simulations1 practice question2 Python programs4 problems1 Python lab
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Lesson 3about 30 min
Orthogonal trajectories and curve problems
Two families are orthogonal trajectories of each other when every curve of one crosses every curve of the other at right angles. To find them, form the differential equation of the first family and replace $\dfrac{dy}{dx}$ by $-\dfrac{dx}{dy}$ . Curve problems turn words about tangents, normals and intercepts into a differential equation.
1 simulation1 practice question1 Python program3 problems1 question set1 note
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Modelling with Differential Equations: Chapter Test

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Modelling with Differential Equations

Course outline (3 lessons)

Growth and decay

Cooling and mixing

Orthogonal trajectories and curve problems

Test Your Learning

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