Coupled First Order Linear Equations (Edexcel A Level Further Maths): Revision Note

Coupled First Order Linear Equations

What are coupled first order linear differential equations?

• Coupled first order linear differential equations are a pair of simultaneous differential equations of the form

  

  • a, b, c and d are real constants

  • f(t) and g(t) are functions of t

    • In your exam these functions will usually be either zero or else simply equal to a constant

  • The equations are described as ‘coupled’ because the rate of change of each of the variables depends not only on the variable itself but also on the other variable

• Systems of coupled differential equations often occur in modelling contexts where two variables are expected to interact

  • For example x may refer to the size of a population of prey animals, and y to the size of a population of predators

    • We would expect the rate of change of the prey animal population to depend on the number of prey animals there are to reproduce, but also on the number of predator animals eating the prey animals

    • Similarly we would expect the rate of change of the predator animal population to depend on the number of predator animals there are to reproduce, but also on the number of prey animals there are for the predators to eat

How do I solve coupled first order linear differential equations?

• You can solve coupled systems by turning them into an uncoupled second order differential equation that you know how to solve

  • For example, consider the coupled system

  • 

• STEP 1:  Rearrange one of the equations to make the variable that is not in the derivative the subject

  • We can rearrange the first equation to get 

• STEP 2:  Differentiate both sides of the equation from Step 1 with respect to t

  • Differentiating gives 

• STEP 3:  Substitute the equations from Steps 1 and 2 into the coupled differential equation you didn’t use in Step 1

  • Substituting into the second equation gives 

    

  • The result is a second order differential equation in only one of the variables

• STEP 4:  Solve the second order differential equation resulting from Step 3

  • Using the standard solution methods for second order differential equations gives the solution 

• STEP 5:  To find the solution for the other variable, substitute the solution from Step 4 along with its derivative into the equation from Step 1

  • Differentiate    to get 

  • Then substituting into the Step 1 equation gives

• STEP 6:  If the question provides initial or boundary conditions you may use these to find the values of A and B, and then go on to interpret your solutions in the context of whatever situation the coupled system of differential equations may be modelling