Coupled Differential Equations (DP IB Maths: AI HL)

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Solving Coupled Differential Equations

How do I write a system of coupled differential equations in matrix form?

  • The coupled differential equations considered in this part of the course will be of the form

fraction numerator d x over denominator d t end fraction equals a x plus b y

fraction numerator d y over denominator d t end fraction equals c x plus d y

    •  a comma space b comma space c comma space d space element of space straight real numbersare constants whose precise value will depend on the situation being modelled
      • In an exam question the values of the constants will generally be given to you
  • This system of equations can also be represented in matrix form:

open parentheses table row cell fraction numerator d x over denominator d t end fraction end cell row cell fraction numerator d y over denominator d t end fraction end cell end table close parentheses equals open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses

  • It is usually more convenient, however, to use the ‘dot notation’ for the derivatives:

open parentheses table row cell x with dot on top end cell row cell y with dot on top end cell end table close parentheses equals open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses

  • This can be written even more succinctly as bold italic x with bold dot on top bold equals bold italic M bold italic x
    • Here bold italic x with bold dot on top equals open parentheses table row cell x with dot on top end cell row cell y with dot on top end cell end table close parentheses, bold italic M equals open parentheses table row a b row c d end table close parentheses, and bold italic x equals open parentheses table row x row y end table close parentheses

How do I find the exact solution for a system of coupled differential equations?

  • The exact solution of the coupled system bold x with bold dot on top equals bold italic M bold italic x depends on the eigenvalues and eigenvectors of the matrix of coefficients bold italic M equals open parentheses table row a b row c d end table close parentheses
    • The eigenvalues and/or eigenvectors may be given to you in an exam question
    • If they are not then you will need to calculate them using the methods learned in the matrices section of the course
  • On the exam you will only be asked to find exact solutions for cases where the two eigenvalues of the matrix are real, distinct, and non-zero
    • Similar solution methods exist for non-real, non-distinct and/or non-zero eigenvalues, but you don’t need to know them as part of the IB AI HL course
  • Let the eigenvalues and corresponding eigenvectors of matrix bold italic M be lambda subscript 1 and lambda subscript 2, and bold italic p subscript 1 and bold italic p subscript 2, respectively
    • Remember from the definition of eigenvalues and eigenvectors that this means that bold italic M bold italic p subscript 1 equals lambda subscript 1 bold italic p subscript 1 and bold italic M bold italic p subscript 2 equals lambda subscript 2 bold italic p subscript 2
  • The exact solution to the system of coupled differential equations is then

bold italic x equals A straight e to the power of lambda subscript 1 t end exponent bold italic p subscript 1 plus B straight e to the power of lambda subscript 2 t end exponent bold italic p subscript bold 2

    • This solution formula is in the exam formula booklet
    • A comma space B space element of space straight real numbers are constants (they are essentially constants of integration of the sort you have when solving other forms of differential equation)
  • If initial or boundary conditions have been provided you can use these to find the precise values of the constants A and B
    • Finding the values of A and B will generally involve solving a set of simultaneous linear equations (see the worked example below)

Worked example

The rates of change of two variables, x and y, are described by the following system of coupled differential equations:

fraction numerator d x over denominator d t end fraction equals 4 x minus y
fraction numerator d y over denominator d t end fraction equals 2 x plus y  

Initially x equals 2 and y equals 1.

 

Given that the matrix open parentheses table row 4 cell negative 1 end cell row 2 1 end table close parentheses has eigenvalues of 3 and 2 with corresponding eigenvectors open parentheses table row 1 row 1 end table close parentheses and open parentheses table row 1 row 2 end table close parentheses, find the exact solution to the system of coupled differential equations.

5-7-1-ib-ai-hl-solving-coupled-diff-eqns-we-solution

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Phase Portraits

What is a phase portrait for a system of coupled differential equations?

  • Here we are again considering systems of coupled equations that can be represented in the matrix form bold italic x with bold dot on top equals bold italic M bold italic x, where bold italic x with bold dot on top equals open parentheses table row cell x with dot on top end cell row cell y with dot on top end cell end table close parentheses, bold italic M equals open parentheses table row a b row c d end table close parentheses, and bold italic x equals open parentheses table row x row y end table close parentheses
  • A phase portrait is a diagram showing how the values of x and y change over time
    • On a phase portrait we will usually sketch several typical solution trajectories
    • The precise trajectory that the solution for a particular system will travel along is determined by the initial conditions for the system
  • Let lambda subscript 1 and lambda subscript 2 be the eigenvalues of the matrix bold italic M
    • The overall nature of the phase portrait depends in large part on the values of lambda subscript 1 and lambda subscript 2

What does the phase portrait look like when lambda subscript 1 and lambda subscript 2 are real numbers?

  • Recall that for real distinct eigenvalues the solution to a system of the above form is bold italic x equals A straight e to the power of lambda subscript 1 t end exponent bold italic p subscript 1 bold plus B straight e to the power of lambda subscript 2 t end exponent bold italic p subscript bold 2, where lambda subscript 1 and lambda subscript 2 are the eigenvalues of bold italic M and bold italic p subscript 1 and bold italic p subscript 2 are the corresponding eigenvectors
    • AI HL only considers cases where lambda subscript 1 and lambda subscript 2 are distinct (i.e., lambda subscript 1 not equal to lambda subscript 2) and non-zero
  • A phase portrait will always include two ‘eigenvector lines’ through the origin, each one parallel to one of the eigenvectors
      • So if bold italic p subscript 1 equals open parentheses table row 1 row 2 end table close parentheses and bold italic p subscript 2 equals open parentheses table row cell negative 3 end cell row 4 end table close parentheses, for example, then these lines through the origin will have equations y equals 2 x and y equals negative 4 over 3 x, respectively
    • These lines will define two sets of solution trajectories
    • If the eigenvalue corresponding to a line’s eigenvector is positive, then there will be solution trajectories along the line away from the origin in both directions as t increases
    • If the eigenvalue corresponding to a line’s eigenvector is negative, then there will be solution trajectories along the line towards the origin in both directions as t increases
    • No solution trajectory will ever cross an eigenvector line
  • If both eigenvalues are positive then all solution trajectories will be directed away from the origin as t increases
    • In between the ‘eigenvector lines’ the trajectories as they move away from the origin will all curve to become approximately parallel to the line whose eigenvector corresponds to the larger eigenvalue

5-7-1-ib-ai-hl-real-pos-pos-eigenvalues-1

  • If both eigenvalues are negative then all solution trajectories will be directed towards the origin as t increases
    • In between the ‘eigenvector lines’ the trajectories will all curve so that at points further away from the origin they are approximately parallel to the line whose eigenvector corresponds to the more negative eigenvalue
      • They will then converge on the other eigenvalue line as they move in towards the origin

5-7-1-ib-ai-hl-real-neg-neg-eigenvalues-1

  • If one eigenvalue is positive and one eigenvalue is negative then solution trajectories will generally start by heading in towards the origin before curving to head out away again from the origin as t increases
    • In between the ‘eigenvector lines’ the solution trajectories will all move in towards the origin along the direction of the eigenvector line that corresponds to the negative eigenvalue, before curving away and converging on the eigenvector line that corresponds to the positive eigenvalue as they head away from the origin

5-7-1-ib-ai-hl-real-pos-neg-eigenvalues

What does the phase portrait look like when lambda subscript 1 and lambda subscript 2 are imaginary numbers?

  • Here the solution trajectories will all be either circles or ellipses with their centres at the origin

5-7-1-ib-ai-hl-imaginary-eigenvalues

  • You can determine the direction (clockwise or anticlockwise) and the shape (circular or elliptical) of the trajectories by considering the values of fraction numerator d x over denominator d t end fraction and fraction numerator d y over denominator d t end fraction for points on the coordinate axes
    • For example, consider the system bold italic x with bold dot on top equals open parentheses table row 1 cell negative 2 end cell row 1 cell negative 1 end cell end table close parentheses bold italic x
      • The eigenvalues of open parentheses table row 1 cell negative 2 end cell row 1 cell negative 1 end cell end table close parentheses are straight i and negative straight i, so the trajectories will be elliptical or circular
    • When ­x equals 1 and y equals 0, fraction numerator d x over denominator d t end fraction equals 1 left parenthesis 1 right parenthesis minus 2 left parenthesis 0 right parenthesis equals 1 and fraction numerator d y over denominator d t end fraction equals 1 left parenthesis 1 right parenthesis minus 1 left parenthesis 0 right parenthesis equals 1
      • This shows that from a point on the positive x-axis the solution trajectory will be moving ‘to the right and up’ in the direction of the vector open parentheses table row 1 row 1 end table close parentheses
    • When ­x equals 0 and y equals 1, fraction numerator d x over denominator d t end fraction equals 1 left parenthesis 0 right parenthesis minus 2 left parenthesis 1 right parenthesis equals negative 2 and fraction numerator d y over denominator d t end fraction equals 1 left parenthesis 0 right parenthesis minus 1 left parenthesis 1 right parenthesis equals negative 1
      • This shows that from a point on the positive y-axis the solution trajectory will be moving ‘to the left and down’ in the direction of the vector open parentheses table row cell negative 2 end cell row cell negative 1 end cell end table close parentheses
    • The directions of the trajectories at those points tell us that the directions of the trajectories will be anticlockwise
    • They also tell us that the trajectories will be ellipses
      • For circular trajectories, the direction of the trajectories when they cross a coordinate axis will be perpendicular to that coordinate axis

What does the phase portrait look like when lambda subscript 1 and lambda subscript 2 are complex numbers?

  • In this case lambda subscript 1 and lambda subscript 2 will be complex conjugates of the form a plus-or-minus b straight i, where a and b are non-zero real numbers
    • If a equals 0, b not equal to 0, then we have the imaginary eigenvalues case above
  • Here the solution trajectories will all be spirals
    • If the real part of the eigenvalues is positive (i.e., if a greater than 0), then the trajectories will spiral away from the origin
    • If the real part of the eigenvalues is negative (i.e., if a less than 0), then the trajectories will spiral towards the origin

5-7-1-ib-ai-hl-complex-eigenvalues

  • You can determine the direction (clockwise or anticlockwise) of the trajectories  by considering the values of fraction numerator d x over denominator d t end fraction and fraction numerator d y over denominator d t end fraction for points on the coordinate axes
    • For example, consider the system bold italic x with dot on top equals open parentheses table row 1 5 row cell negative 2 end cell 3 end table close parentheses bold italic x
      • The eigenvalues of open parentheses table row 1 cell negative 5 end cell row 2 3 end table close parentheses are 2 plus 3 straight i and 2 minus 3 straight i, so the trajectories will be spirals
      • Because the real part of the eigenvalues left parenthesis 2 right parenthesis is positive, the trajectories will spiral away from the origin
    • When ­x equals 1 and y equals 0, fraction numerator d x over denominator d t end fraction equals 1 left parenthesis 1 right parenthesis plus 5 left parenthesis 0 right parenthesis equals 1 and fraction numerator d y over denominator d t end fraction equals negative 2 left parenthesis 1 right parenthesis plus 3 left parenthesis 0 right parenthesis equals negative 2
      • This shows that from a point on the positive x-axis the solution trajectory will be moving ‘to the right and down’ in the direction of the vector open parentheses table row 1 row cell negative 2 end cell end table close parentheses
    • When ­x equals 0 and y equals 1, fraction numerator d x over denominator d t end fraction equals 1 left parenthesis 0 right parenthesis plus 5 left parenthesis 1 right parenthesis equals 5 and fraction numerator d t over denominator d t end fraction equals negative 2 left parenthesis 0 right parenthesis plus 3 left parenthesis 1 right parenthesis equals 3
      • This shows that from a point on the positive y-axis the solution trajectory will be moving ‘to the right and up’ in the direction of the vector open parentheses table row 5 row 3 end table close parentheses
    • The directions of the trajectories at those points tell us that the directions of the trajectory spirals will be clockwise

Worked example

Consider the system of coupled differential equations

 fraction numerator d x over denominator d t end fraction equals negative 2 x plus 2 y
fraction numerator d y over denominator d t end fraction equals x minus 3 y

Given that negative 1 and negative 4 are the eigenvalues of the matrix open parentheses table row cell negative 2 end cell 2 row 1 cell negative 3 end cell end table close parentheses, with corresponding eigenvectors open parentheses table row 2 row 1 end table close parentheses and open parentheses table row cell negative 1 end cell row 1 end table close parentheses, draw a phase portrait for the solutions of the system.

5-7-1-ib-ai-hl-phase-portraits-we-solution

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Equilibrium Points

What is an equilibrium point?

  • For a system of coupled differential equations, an equilibrium point is a point left parenthesis x comma space y right parenthesis at which both fraction numerator d x over denominator d t end fraction equals 0 and fraction numerator d y over denominator d t end fraction equals 0
    • Because both derivatives are zero, the rates of change of both x and y are zero
    • This means that x and y will not change, and therefore that if the system is ever at the point left parenthesis x comma space y right parenthesis then it will remain at that point left parenthesis x comma space y right parenthesis forever
  • An equilibrium point can be stable or unstable
    • An equilibrium point is stable if for all points close to the equilibrium point the solution trajectories move back towards the equilibrium point
      • This means that if the system is perturbed away from the equilibrium point, it will tend to move back towards the state of equilibrium
    • If an equilibrium point is not stable, then it is unstable
      • If a system is perturbed away from an unstable equilibrium point, it will tend to continue moving further and further away from the state of equilibrium
  • For a system that can be represented in the matrix form bold italic x with dot on top equals bold italic M bold italic x, where bold italic x with bold dot on top equals open parentheses table row cell x with dot on top end cell row cell y with dot on top end cell end table close parentheses, bold italic M equals open parentheses table row a b row c d end table close parentheses, and bold italic x equals open parentheses table row x row y end table close parentheses, the origin left parenthesis 0 comma space 0 right parenthesis is always an equilibrium point
    • Considering the nature of the phase portrait for a particular system will tell us what sort of equilibrium point the origin is
    • If both eigenvalues of the matrix bold italic M are real and negative, then the origin is a stable equilibrium point
      • This sort of equilibrium point is sometimes known as a sink
    • If both eigenvalues of the matrix bold italic M are real and positive, then the origin is an unstable equilibrium point
      • This sort of equilibrium point is sometimes known as a source
    • If both eigenvalues of the matrix bold italic M are real, with one positive and one negative, then the origin is an unstable equilibrium point
      • This sort of equilibrium point is known as a saddle point (you will be expected to identify saddle points if they occur in an AI HL exam question)
    • If both eigenvalues of the matrix bold italic M are imaginary, then the origin is an unstable equilibrium point
      • Recall that for all points other than the origin, the solution trajectories here all ‘orbit’ around the origin along circular or elliptical paths
    • If both eigenvalues of the matrix bold italic M are complex with a negative real part, then the origin is an stable equilibrium point
      • All solution trajectories here spiral in towards the origin
    • If both eigenvalues of the matrix bold italic M are complex with a positive real part, then the origin is an unstable equilibrium point
      • All solution trajectories here spiral away from the origin

Worked example

a)
Consider the system of coupled differential equations

 fraction numerator d x over denominator d t end fraction equals 2 x minus 3 y plus 6
fraction numerator d y over denominator d t end fraction equals x plus y minus 7

Show that left parenthesis 3 comma space 4 right parenthesis is an equilibrium point for the system.

5-7-1-ib-ai-hl-equilibrium-points-a-we-solution

b)
Consider the system of coupled differential equations

fraction numerator d x over denominator d t end fraction equals x plus 3 y
fraction numerator d y over denominator d t end fraction equals 2 x plus 2 y

Given that 4 and negative 1 are the eigenvalues of the matrix open parentheses table row 1 3 row 2 2 end table close parentheses, with corresponding eigenvectors open parentheses table row 1 row 1 end table close parentheses and open parentheses table row cell negative 3 end cell row 2 end table close parentheses, determine the coordinates and nature of the equilibrium point for the system.

5-7-1-ib-ai-hl-equilibrium-points-b-we-solution

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Sketching Solution Trajectories

How do I sketch a solution trajectory for a system of coupled differential equations?

  • A phase portrait shows typical trajectories representing all the possible solutions to a system of coupled differential equations
  • For a given set of initial conditions, however, the solution will only have one specific trajectory
  • Sketching a particular solution trajectory will generally involve the following:
    • Make sure you know what the ‘typical’ solutions for the system look like
      • You don’t need to sketch a complete phase portrait unless asked, but you should know what the phase portrait for your system would look like
      • If the phase portrait includes ‘eigenvector lines’, however, it is worth including these in your sketch to serve as guidelines
    • Mark the starting point for your solution trajectory
      • The coordinates of the starting point will be the x and y values when t equals 0
      • Usually these are given in the question as the initial conditions for the system
    • Determine the initial direction of the solution trajectory
      • To do this find the values of fraction numerator d x over denominator d t end fraction and fraction numerator d y over denominator d t end fraction when t equals 0
      • This will tell you the directions in which x and y are changing initially
      • For example if fraction numerator d x over denominator d t end fraction equals negative 2 and fraction numerator d y over denominator d t end fraction equals 3 when t equals 0, then the trajectory from the starting point will initially be ‘to the left and up’, parallel to the vector open parentheses table row cell negative 2 end cell row 3 end table close parentheses
    • Use the above considerations to create your sketch
      • The trajectory should begin at the starting point (be sure to mark and label the starting point on your sketch!)
      • It should move away from the starting point in the correct initial direction
      • As it moves further away from the starting point, the trajectory should conform to the nature of a ‘typical solution’ for the system

Worked example

Consider the system of coupled differential equations

fraction numerator d x over denominator d t end fraction equals x minus 5 y
fraction numerator d y over denominator d t end fraction equals negative 3 x plus 3 y

The initial conditions of the system are such that the exact solution is given by 

bold italic x equals straight e to the power of 6 t end exponent open parentheses table row cell negative 1 end cell row 1 end table close parentheses minus 2 straight e to the power of negative 2 t end exponent open parentheses table row 5 row 3 end table close parentheses

Sketch the trajectory of the solution, showing the relationship between x and y as t increases from zero.

IVk5AY-f_5-7-1-ib-ai-hl-sketching-solution-trajectories-we-solution

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.