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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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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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30. More partial differential equations
- Class notes • 5 pages • 2019
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Bessel equation and looking at Frobenius solution when using cylindrical coordinates. Show how nu is a solution a Frobenius solution to the Bessel Equation. Finding a recurrence relation for the coefficients of the Bessel functions. Plotting Bessel functions of the first kind. Generalising the Bessel series solution of first kind of order nu where half integers can used.
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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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Exam Revision
- Summary • 14 pages • 2019
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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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22. PDEs and waves
- Class notes • 5 pages • 2019
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Matlab Script graphing Wave Equation solution of separation of variables with Fourier series. Heat Flow equation example solving and graphing with Matlab. PDEs and waves, shows part way through deriving the PDE for the wave equation.
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7. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Example finished from the end of the last notes. Definitions of linearly independent, linear dependence and linear combination with an example. Existence and uniqueness theorem. Example of solving a third order IVP. Definition of the Wronskian, the Wronskian test with examples and a proof.
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9. Second and higher order ODEs
- Class notes • 5 pages • 2019
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Existence and completeness theorem and a general proof (not complete), Characteristic Equation derivation and the different types with an example. Theory of linear independent solutions and a basis, using Wronksian for linear independence and basis- partial proof. Complex roots theory and an example with a graph and matlab code. Multiple roots theory of the number of linearly independent solutions and an example.
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11. Second and higher order ODEs
- Class notes • 5 pages • 2019
- Available in package deal
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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.