coupled-first-order-linear-equations

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 <img alt=”space y equals 0.5 fraction numerator d x over denominator d t end fraction minus 0.3 x” data-mathml='<math ><semantics><mrow><mo> </mo><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>x</mi></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”47″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2247%22%20width%3D%22133%22%20wrs%3Abaseline%3D%2230%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E.%3C%2Fmo%3E%3Cmn%3E5%3C%2Fmn%3E%3Cmfrac%3E%3Cmrow%3E%3Cmo%3