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10. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Proof for multiple roots for solutions. Theory for the general solution for nonhomogeneous linear ODEs and an example. Method of undetermined guessing and an example which is plotted with matlab code. An example of an IVP for nonhomogeneous solutions.
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11. Second and higher order ODEs
- Class notes • 5 pages • 2019
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General solution for two nonhomogeneous ODE example. How to use matlab to check your solutions. Modelling free and forced oscillations theory and finding the solutions to an undamped system. Example of a harmonic oscillator and looking at the different types of damped systems.
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12. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Damped system and the different scenarios and their solutions. Solutions plotted in matlab and graphed. Looked at Resonance in forced oscillations and the particular solution for this equation. Graph the solution of types of solution for the particular solution for different constant values and interpreted the results. Looked at more general methods of solving ODEs: finding a solution that is a factor multiple of a known solution.
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13. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Reduction of order method to find all terms of the homogeneous solution. Euler Cauchy Equations and the types of solution for the roots of the characteristic equation. Example using Reduction of order finding the general solution. Modelling Mathematical ecology, by looking at a fish example, finding solutions to a system of equations of how different factors affect fish populations.
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14. Second and higher order ODEs
- Class notes • 4 pages • 2019
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Looking at steady state solution for modelling the populations of predators and prey with graphs. Constructing systems of equations for different stages of life to model the population of a species.
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15. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Solving systems of equations using matlab and creating a general solution. Shows example of a repeated root problem and how to solve. Goes through the theorem for system of equations and a proof. Gives an example of modelling a system of equations.
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16. Partial differential equations
- Class notes • 4 pages • 2019
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Partial Differential Equations and its definition and properties. Some important PDEs are given and an example given of how to solve one. It goes through how to make a well-posed PDE question and the modelling framework. An example is given of modelling car traffic going through the steps of the modelling framework.
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17. Partial differential equations
- Class notes • 5 pages • 2019
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Example of Near constant car density continued with interpretation. D'Alembert's solutions with an example of some cars being able to pass other cars. Heat equation and solving it with separation of variables coming to the full general solution.
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18. Representing periodic functions by Fourier series
- Class notes • 3 pages • 2019
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Square Wave example with matlab script to plot solution. Theorem of functions being orthogonal and proving this is the case for different periods. Deriving the value of the coefficients for the fourier series. Square wave Redux.
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19. Representing periodic functions by Fourier series
- Class notes • 5 pages • 2019
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Separation of variables to solve the Heat equation and represent as a summation solution. Example of copper bar and solving for its solution to transfer of heat over time. Includes matlab code of graphing of solution. Different Boundary conditions: Dirichlet and Neumann boundary conditions, motivation for Fourier series.
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