Modelling using First Order Differential Equations

Why are differential equations used to model real-world situations?

A differential equation is an equation that contains one or more derivatives
Derivatives deal with rates of change, and with the way that variables change with respect to one another
Therefore differential equations are a natural way to model real-world situations involving change

Most frequently in real-world situations we are interested in how things change over time, so the derivatives used will usually be with respect to time t

How do I set up a differential equation to model a situation?

An exam question may require you to create a differential equation from information provided
The question will provide a context from which the differential equation is to be created
Most often this will involve the rate of change of a variable being proportional to some function of the variable
- - For example, the rate of change of a population of bacteria, P, at a particular time may be proportional to the size of the population at that time
The expression ‘rate of’ (‘rate of change of…’, ‘rate of growth of…’, etc.) in a modelling question is a strong hint that a differential equation is needed, involving derivatives with respect to time t
- - So with the bacteria example above, the equation will involve the derivative $\frac{d P}{d t}$
Recall the basic equation of proportionality
- If y is proportional to x, then y = kx for some constant of proportionality k
  - So for the bacteria example above the differential equation needed would be $\frac{d P}{d t} = k P$
- The precise value of k will generally not be known at the start, but will need to be found as part of the process of solving the differential equation
- It can often be useful to assume that k > 0 when setting up your equation
  - In this case, -k will be used in the differential equation in situations where the rate of change is expected to be negative
  - So in the bacteria example, if it were known that the population of bacteria was decreasing, then the equation could instead be written $\frac{d P}{d t} = - k P$
Often scenarios will involves multiple things than affect the rate
- If something causes the variable to increase then that term will be added to the rate of change
  - Such as water flowing into a space
- If something causes the variable to decrease then that term will be subtracted from the rate of change
  - Such as water flowing out of space

Worked example

In a particular pond, the rate of change of the area covered by algae, A, at any time t is directly proportional to the square root of the area covered by algae at that time. Write down a differential equation to model this situation.

5-10-3-ib-aa-hl-modelling-with-diff-eqns-a-we-solution

Newton’s Law of Cooling states that the rate of change of the temperature of an object, T, at any time t is proportional to the difference between the temperature of the object and the ambient temperature of its surroundings, T_a , at that time. Assuming that the object starts off warmer than its surroundings, write down the differential equation implied by Newton’s Law of Cooling.

5-10-3-ib-aa-hl-modelling-with-diff-eqns-b-we-solution

Modelling using First Order Differential Equations (Edexcel A Level Further Maths: Core Pure)

Revision Note

Why are differential equations used to model real-world situations?

How do I set up a differential equation to model a situation?

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1. Complex Numbers

1.1 Complex Numbers & Argand Diagrams

1.1.1 Introduction to Complex Numbers

1.1.2 Solving Equations with Complex Roots

1.1.3 Modulus & Argument

1.1.4 Modulus-Argument Form

1.1.5 Loci in Argand Diagrams

1.1.6 Regions in Argand Diagrams

1.2 Exponential Form & de Moivre's Theorem

1.2.1 Exponential Form

1.2.2 de Moivre's Theorem

1.2.3 Applications of de Moivre's Theorem

1.2.4 Roots of Complex Numbers

2. Matrices

2.1 Properties of Matrices

2.1.1 Introduction to Matrices

2.1.2 Determinants of Matrices

2.1.3 Inverses of Matrices

2.2 Transformations using Matrices

2.2.1 Transformations using a Matrix

2.2.2 Geometric Transformations with Matrices

2.2.3 Invariant Points & Lines

3. Further Algebra & Functions

3.1 Roots of Polynomials

3.1.1 Roots of Polynomials

3.1.2 Linear Transformations of Roots

3.2 Series

3.2.1 Sums of Integers, Squares & Cubes

3.2.2 Method of Differences

3.3 Maclaurin Series

3.3.1 Maclaurin Series

4. Hyperbolic Functions

4.1 Hyperbolic Functions

4.1.1 Hyperbolic Functions & Graphs

4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

4.1.3 Hyperbolic Identities & Equations

4.1.4 Differentiating & Integrating Hyperbolic Functions

5. Further Calculus

5.1 Volumes of Revolution

5.1.1 Volumes of Revolution

5.1.2 Modelling with Volumes of Revolution

5.2 Methods in Calculus

5.2.1 Improper Integrals

5.2.2 Mean Value of a Function

5.2.3 Integrating with Partial Fractions

5.2.4 Calculus involving Inverse Trig

5.2.5 Integration by Substitution

6. Further Vectors

6.1 Vector Lines

6.1.1 Equations of Lines in 3D

6.1.2 Pairs of Lines in 3D

6.1.3 Angle between Lines

6.1.4 Shortest Distances - Lines

6.2 Vector Planes

6.2.1 Equations of planes

6.2.2 Combinations of Lines & Planes

6.2.3 Combinations of Planes

6.2.4 Shortest Distances - Planes

7. Polar Coordinates

7.1 Polar Coordinates

7.1.1 Polar Coordinates

7.1.2 Calculus with Polar Coordinates

8. Differential Equations

8.1 First Order Differential Equations

8.1.1 Intro to Differential Equations

8.1.2 Solving First Order Differential Equations

8.1.3 Modelling using First Order Differential Equations

8.2 Second Order Differential Equations

8.2.1 Solving Second Order Differential Equations

8.2.2 Coupled First Order Linear Equations

8.3 Simple Harmonic Motion

8.3.1 Simple Harmonic Motion

8.3.2 Damped or Forced Harmonic Motion