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Revision Notes

Home / IB / Maths: AI HL / DP / Revision Notes / 5. Calculus / 5.7 Further Differential Equations / 5.7.2 Second Order Differential Equations


5.7.2 Second Order Differential Equations


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 right parenthesis 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

 

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



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Approximation & Estimation
          • 1.1.3 GDC: Solving Equations
          • 1.2 Exponentials & Logs
            • 1.2.1 Exponents
              • 1.2.2 Logarithms
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                      • 1.4 Financial Applications
                        • 1.4.1 Compound Interest & Depreciation
                          • 1.4.2 Amortisation & Annuities
                          • 1.5 Complex Numbers
                            • 1.5.1 Intro to Complex Numbers
                              • 1.5.2 Modulus & Argument
                                • 1.5.3 Introduction to Argand Diagrams
                                • 1.6 Further Complex Numbers
                                  • 1.6.1 Geometry of Complex Numbers
                                    • 1.6.2 Forms of Complex Numbers
                                      • 1.6.3 Applications of Complex Numbers
                                      • 1.7 Matrices
                                        • 1.7.1 Introduction to Matrices
                                          • 1.7.2 Operations with Matrices
                                            • 1.7.3 Determinants & Inverses
                                              • 1.7.4 Solving Systems of Linear Equations with Matrices
                                              • 1.8 Eigenvalues & Eigenvectors
                                                • 1.8.1 Eigenvalues & Eigenvectors
                                                  • 1.8.2 Applications of Matrices
                                                • 2. Functions
                                                  • 2.1 Linear Functions & Graphs
                                                    • 2.1.1 Equations of a Straight Line
                                                    • 2.2 Further Functions & Graphs
                                                      • 2.2.1 Functions
                                                        • 2.2.2 Graphing Functions
                                                          • 2.2.3 Properties of Graphs
                                                          • 2.3 Modelling with Functions
                                                            • 2.3.1 Linear Models
                                                              • 2.3.2 Quadratic & Cubic Models
                                                                • 2.3.3 Exponential Models
                                                                  • 2.3.4 Direct & Inverse Variation
                                                                    • 2.3.5 Sinusoidal Models
                                                                      • 2.3.6 Strategy for Modelling Functions
                                                                      • 2.4 Functions Toolkit
                                                                        • 2.4.1 Composite & Inverse Functions
                                                                        • 2.5 Transformations of Graphs
                                                                          • 2.5.1 Translations of Graphs
                                                                            • 2.5.2 Reflections of Graphs
                                                                              • 2.5.3 Stretches of Graphs
                                                                                • 2.5.4 Composite Transformations of Graphs
                                                                                • 2.6 Further Modelling with Functions
                                                                                  • 2.6.1 Properties of Further Graphs
                                                                                    • 2.6.2 Natural Logarithmic Models
                                                                                      • 2.6.3 Logistic Models
                                                                                        • 2.6.4 Piecewise Models
                                                                                      • 3. Geometry & Trigonometry
                                                                                        • 3.1 Geometry Toolkit
                                                                                          • 3.1.1 Coordinate Geometry
                                                                                            • 3.1.2 Radian Measure
                                                                                              • 3.1.3 Arcs & Sectors
                                                                                              • 3.2 Geometry of 3D Shapes
                                                                                                • 3.2.1 3D Coordinate Geometry
                                                                                                  • 3.2.2 Volume & Surface Area
                                                                                                  • 3.3 Trigonometry
                                                                                                    • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                      • 3.3.2 Non Right-Angled Trigonometry
                                                                                                        • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                        • 3.4 Further Trigonometry
                                                                                                          • 3.4.1 The Unit Circle
                                                                                                            • 3.4.2 Simple Identities
                                                                                                              • 3.4.3 Solving Trigonometric Equations
                                                                                                              • 3.5 Voronoi Diagrams
                                                                                                                • 3.5.1 Voronoi Diagrams
                                                                                                                  • 3.5.2 Toxic Waste Dump Problem
                                                                                                                  • 3.6 Matrix Transformations
                                                                                                                    • 3.6.1 Matrix Transformations
                                                                                                                      • 3.6.2 Determinant of a Transformation Matrix
                                                                                                                      • 3.7 Vector Properties
                                                                                                                        • 3.7.1 Introduction to Vectors
                                                                                                                          • 3.7.2 Position & Displacement Vectors
                                                                                                                            • 3.7.3 Magnitude of a Vector
                                                                                                                              • 3.7.4 The Scalar Product
                                                                                                                                • 3.7.5 The Vector Product
                                                                                                                                  • 3.7.6 Components of Vectors
                                                                                                                                    • 3.7.7 Geometric Proof with Vectors
                                                                                                                                    • 3.8 Vector Equations of Lines
                                                                                                                                      • 3.8.1 Vector Equations of Lines
                                                                                                                                        • 3.8.2 Shortest Distances with Lines
                                                                                                                                        • 3.9 Modelling with Vectors
                                                                                                                                          • 3.9.1 Kinematics with Vectors
                                                                                                                                            • 3.9.2 Constant & Variable Velocity
                                                                                                                                            • 3.10 Graph Theory
                                                                                                                                              • 3.10.1 Introduction to Graph Theory
                                                                                                                                                • 3.10.2 Walks & Adjacency Matrices
                                                                                                                                                  • 3.10.3 Minimum Spanning Trees
                                                                                                                                                    • 3.10.4 Chinese Postman Problem
                                                                                                                                                      • 3.10.5 Travelling Salesman Problem
                                                                                                                                                        • 3.10.6 Bounds for Travelling Salesman Problem
                                                                                                                                                      • 4. Statistics & Probability
                                                                                                                                                        • 4.1 Statistics Toolkit
                                                                                                                                                          • 4.1.1 Sampling
                                                                                                                                                            • 4.1.2 Data Collection
                                                                                                                                                              • 4.1.3 Statistical Measures
                                                                                                                                                                • 4.1.4 Frequency Tables
                                                                                                                                                                  • 4.1.5 Linear Transformations of Data
                                                                                                                                                                    • 4.1.6 Outliers
                                                                                                                                                                      • 4.1.7 Univariate Data
                                                                                                                                                                        • 4.1.8 Interpreting Data
                                                                                                                                                                        • 4.2 Correlation & Regression
                                                                                                                                                                          • 4.2.1 Bivariate Data
                                                                                                                                                                            • 4.2.2 Correlation Coefficients
                                                                                                                                                                              • 4.2.3 Linear Regression
                                                                                                                                                                              • 4.3 Further Correlation & Regression
                                                                                                                                                                                • 4.3.1 Non-linear Regression
                                                                                                                                                                                  • 4.3.2 Logarithmic Scales
                                                                                                                                                                                    • 4.3.3 Linearising using Logarithms
                                                                                                                                                                                    • 4.4 Probability
                                                                                                                                                                                      • 4.4.1 Probability & Types of Events
                                                                                                                                                                                        • 4.4.2 Conditional Probability
                                                                                                                                                                                          • 4.4.3 Sample Space Diagrams
                                                                                                                                                                                          • 4.5 Probability Distributions
                                                                                                                                                                                            • 4.5.1 Discrete Probability Distributions
                                                                                                                                                                                              • 4.5.2 Expected Values
                                                                                                                                                                                              • 4.6 Random Variables
                                                                                                                                                                                                • 4.6.1 Linear Combinations of Random Variables
                                                                                                                                                                                                  • 4.6.2 Unbiased Estimates
                                                                                                                                                                                                  • 4.7 Binomial Distribution
                                                                                                                                                                                                    • 4.7.1 The Binomial Distribution
                                                                                                                                                                                                      • 4.7.2 Calculating Binomial Probabilities
                                                                                                                                                                                                      • 4.8 Normal Distribution
                                                                                                                                                                                                        • 4.8.1 The Normal Distribution
                                                                                                                                                                                                          • 4.8.2 Calculations with Normal Distribution
                                                                                                                                                                                                          • 4.9 Further Normal Distribution (inc Central Limit Theorem)
                                                                                                                                                                                                            • 4.9.1 Sample Mean Distribution
                                                                                                                                                                                                              • 4.9.2 Confidence Interval for the Mean
                                                                                                                                                                                                              • 4.10 Poisson Distribution
                                                                                                                                                                                                                • 4.10.1 Poisson Distribution
                                                                                                                                                                                                                  • 4.10.2 Calculating Poisson Probabilities
                                                                                                                                                                                                                  • 4.11 Hypothesis Testing
                                                                                                                                                                                                                    • 4.11.1 Hypothesis Testing
                                                                                                                                                                                                                      • 4.11.2 Chi-squared Test for Independence
                                                                                                                                                                                                                        • 4.11.3 Goodness of Fit Test
                                                                                                                                                                                                                        • 4.12 Further Hypothesis Testing
                                                                                                                                                                                                                          • 4.12.1 Hypothesis Testing for Mean (One Sample)
                                                                                                                                                                                                                            • 4.12.2 Hypothesis Testing for Mean (Two Sample)
                                                                                                                                                                                                                              • 4.12.3 Binomial Hypothesis Testing
                                                                                                                                                                                                                                • 4.12.4 Poisson Hypothesis Testing
                                                                                                                                                                                                                                  • 4.12.5 Hypothesis Testing for Correlation
                                                                                                                                                                                                                                    • 4.12.6 Type I & Type II Errors
                                                                                                                                                                                                                                    • 4.13 Transition Matrices & Markov Chains
                                                                                                                                                                                                                                      • 4.13.1 Markov Chains
                                                                                                                                                                                                                                        • 4.13.2 Transition Matrices
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                              • 5.1.3 Modelling with Differentiation
                                                                                                                                                                                                                                              • 5.2 Further Differentiation
                                                                                                                                                                                                                                                • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                  • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                    • 5.2.3 Related Rates of Change
                                                                                                                                                                                                                                                      • 5.2.4 Second Order Derivatives
                                                                                                                                                                                                                                                        • 5.2.5 Further Applications of Differentiation
                                                                                                                                                                                                                                                          • 5.2.6 Concavity & Points of Inflection
                                                                                                                                                                                                                                                          • 5.3 Integration
                                                                                                                                                                                                                                                            • 5.3.1 Trapezoid Rule: Numerical Integration
                                                                                                                                                                                                                                                              • 5.3.2 Introduction to Integration
                                                                                                                                                                                                                                                                • 5.3.3 Applications of Integration
                                                                                                                                                                                                                                                                • 5.4 Further Integration
                                                                                                                                                                                                                                                                  • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                    • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                      • 5.4.3 Further Applications of Integration
                                                                                                                                                                                                                                                                        • 5.4.4 Volumes of Revolution
                                                                                                                                                                                                                                                                        • 5.5 Kinematics
                                                                                                                                                                                                                                                                          • 5.5.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                            • 5.5.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                            • 5.6 Differential Equations
                                                                                                                                                                                                                                                                              • 5.6.1 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                • 5.6.2 Separation of Variables
                                                                                                                                                                                                                                                                                  • 5.6.3 Slope Fields
                                                                                                                                                                                                                                                                                    • 5.6.4 Approximate Solutions to Differential Equations
                                                                                                                                                                                                                                                                                    • 5.7 Further Differential Equations
                                                                                                                                                                                                                                                                                      • 5.7.1 Coupled Differential Equations
                                                                                                                                                                                                                                                                                        • 5.7.2 Second Order Differential Equations
                                                                                                                                                                                                                                                                                      Roger Bliss

                                                                                                                                                                                                                                                                                      Author: Roger

                                                                                                                                                                                                                                                                                      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.


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