Second Order Differential Equations (DP IB Maths: AI HL)

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Roger

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Euler's Method: Second Order

How do I apply Euler’s method to second order differential equations?

  • A second order differential equation is a differential equation containing one or more second derivatives
  • In this section of the course we consider second order differential equations of the form

fraction numerator straight d squared x over denominator straight d t squared end fraction equals f open parentheses x comma fraction numerator straight d x over denominator straight d t end fraction comma t close parentheses

    • You may need to rearrange the differential equation given to get it in this form
  • In order to apply Euler’s method, use the substitution space y equals fraction numerator d x over denominator d t end fraction to turn the second order differential equation into a pair of coupled first order differential equations
    • If space y equals fraction numerator d x over denominator d t end fraction, then fraction numerator straight d y over denominator straight d t end fraction equals fraction numerator straight d squared x over denominator straight d t squared end fraction
    • This changes the second order differential equation into the coupled system
  • Approximate solutions to this coupled system can then be found using the standard Euler’s method for coupled systems
    • See the notes on this method in the revision note 5.6.4 Approximate Solutions to Differential Equations

fraction numerator d x over denominator d t end fraction equals y
fraction numerator d y over denominator d t end fraction equals f open parentheses x comma space y comma space t close parentheses

Worked example

Consider the second order differential equation fraction numerator straight d squared x over denominator straight d t squared end fraction plus 2 fraction numerator straight d x over denominator straight d t end fraction plus x equals 50 cos invisible function application t.

a)
Show that the equation above can be rewritten as a system of coupled first order differential equations.

5-7-2-ib-ai-hl-eulers-method-second-order-a-we-solution

b)
Initially x equals 2 and fraction numerator d x over denominator d t end fraction equals negative 1. By applying Euler’s method with a step size of 0.1, find approximations for the values of x and fraction numerator d x over denominator d t end fraction when t = 0.5 .

5-7-2-ib-ai-hl-eulers-method-second-order-b-we-solution

 

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Exact Solutions & Phase Portraits: Second Order

How can I find the exact solution for a second order differential equation?

  • In some cases we can apply methods we already know to find the exact solutions for second order differential equations
  • In this section of the course we consider second order differential equations of the form

fraction numerator straight d squared x over denominator straight d t squared end fraction plus a fraction numerator straight d x over denominator straight d t end fraction plus b x equals 0

    •  are constants
  • Use the substitution space y equals fraction numerator d x over denominator d t end fraction to turn the second order differential equation into a pair of coupled first order differential equations
    • If space y equals fraction numerator d x over denominator d t end fraction, then fraction numerator straight d y over denominator straight d t end fraction equals fraction numerator straight d squared x over denominator straight d t squared end fraction
    • This changes the second order differential equation into the coupled system

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d x over denominator d t end fraction end cell equals y row cell fraction numerator d y over denominator d t end fraction end cell equals cell negative b x minus a y end cell end table

    • The coupled system may also be represented in matrix form as

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 0 1 row cell negative b end cell cell negative a end cell end table close parentheses open parentheses table row x row y end table close parentheses

      • In the ‘dot notation’ here  and
    • That can be written even more succinctly as bold italic x with dot on top equals bold italic M bold italic x
      • Here bold italic x with 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 x equals open parentheses table row x row y end table close parentheses, and bold italic M equals open parentheses table row 0 1 row cell negative b end cell cell negative a end cell end table close parentheses
  • Once the original equation has been rewritten in matrix form, the standard method for finding exact solutions of systems of coupled differential equations may be used
    • The solutions will depend on the eigenvalues and eigenvectors of the matrix M
    • For the details of the solution method see the revision note 5.7.1 Coupled Differential Equations
    • Remember that exam questions will only ask for exact solutions for cases where the eigenvalues of M are real and distinct

How can I use phase portraits to investigate the solutions to second order differential equations?

  • Here we are again considering second order differential equations of the form

fraction numerator straight d squared x over denominator straight d t squared end fraction plus a fraction numerator straight d x over denominator straight d t end fraction plus b x equals 0

    •  a & b are real constants
  • As shown above, the substitution space y equals fraction numerator d x over denominator d t end fraction can be used to convert this second order differential equation into a system of coupled first order differential equations of the form bold italic x with dot on top equals bold italic M bold italic x
    • Here bold italic x with 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 x equals open parentheses table row x row y end table close parentheses, and bold italic M equals open parentheses table row 0 1 row cell negative b end cell cell negative a end cell end table close parentheses
  • Once the equation has been rewritten in this form, you may use the standard methods to construct a phase portrait or sketch a solution trajectory for the equation
    • For the details of the phase portrait and solution trajectory methods see the revision note 5.7.1 Coupled Differential Equations
    • When interpreting a phase portrait or solution trajectory sketch, don’t forget that space y equals fraction numerator d x over denominator d t end fraction
      • So if x represents the displacement of a particle, for example, then space y equals fraction numerator d x over denominator d t end fraction will represent the particle’s velocity

Worked example

Consider the second order differential equation fraction numerator straight d squared x over denominator straight d t squared end fraction plus 3 fraction numerator straight d x over denominator straight d t end fraction minus 4 x equals 0. Initially x = 3 and  fraction numerator d x over denominator d t end fraction equals negative 2.

a)
Show that the equation above can be rewritten as a system of coupled first order differential equations.

5-7-2-ib-ai-hl-exact-second-order-a-we-solution

b)
Given that the matrix open parentheses table row 0 1 row 4 cell negative 3 end cell end table close parentheses has eigenvalues of 1 and -4 with corresponding eigenvectors open parentheses table row 1 row 1 end table close parentheses and open parentheses table row cell negative 1 end cell row 4 end table close parentheses, find the exact solution to the second order differential equation.

5-7-2-ib-ai-hl-exact-second-order-b-we-solution

c)
Sketch the trajectory of the solution to the equation on a phase diagram, showing the relationship between x and fraction numerator d x over denominator d t end fraction.

5-7-2-ib-ai-hl-exact-second-order-c-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.