Final Exam Checklist

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By the end of AME 30314, a student should be able to:

1. Identify the following attributes of a differential equation.
   1. whether it is ordinary or partial
   2. whether it is linear or nonlinear
   3. what its order is
   4. if it is linear, whether it is constant or variable coefficient, and
   5. if it is linear, whether it is homogeneous or inhomogeneous.
2. Identify the dependent and independent variables in a differential equation.
3. Identify what solution method(s) can be used to solve a differential equation based on its attributes.
4. Explain why applicable solution methods work and why non-applicable ones will not work.
5. Use the following methods to solve differential equations where applicable:
   1. Assuming exponential solutions
   2. Undetermined coefficients
   3. Variation of parameters
   4. Solving separable equations
   5. Solving exact equations
   6. Using series solutions
   7. Using separation of variables
   8. Using integrating factors if given the integrating factor or directions on how to compute it.
6. Explain in words where and why each of the methods listed above works.
7. For linear first and second order constant coefficient equations explain how changing parameters in the differential equation will change the features of the solution such as:
   1. whether it is stable or unstable
   2. if it is stable, how quickly it decays
   3. whether or not it oscillates
   4. if it oscillates, the frequency of oscillation
   5. if it oscillates, the magnitude of oscillation
   6. if it approaches a steady state value, what that value is.
8. For linear first and second order constant coefficient equations, match equations with plots of solutions without solving each/all equation in detail.
9. For any solution obtained for any differential equation, sketch the solution and sketch how the solution would change is a parameter in the solution changes.
10. Determine the differential equation describing a physical phenomenon described in words in terms of how rates of changes are related to physical properties, i.e., word problems.
11. Given a differential equation, identify and use all solution methods that can be used to solve the problem.
12. For the principles of superposition and linear independence:
    1. explain what they are
    2. explain why they are important and how they are used in various solution methods from this class
    3. identify which solution methods in the course use rely upon them.
13. For homogeneous, constant coefficient, ordinary, linear, second order differential equations compute the characteristic equation, solutions corresponding to various types of roots of the characteristic equation, explain the attributes of the solutions and sketch them.
14. Use undetermined coefficients when the particular solution contains a homogeneous solution.
15. With some guidance, use any of the solution methods on a class of problems not covered in class.
16. Determine and solve the equation of motion for single degree of freedom mass-spring-damper systems in various configurations,
    1. including when the spring and damper elements are attached to fixed and moving bases
    2. including when the applied force is produced by an unbalanced rotating element.
17. Determine how changes in the system such as changes in spring or damper constants, forcing frequency, forcing magnitude, etc. can affect the nature of the motion of the system including the effect on
    1. frequency of oscillation
    2. magnitude of oscillations
    3. magnitudes of various forces in the system including the forces in other elements, on walls, the total force on the mass.
18. Recommend based on analysis changes in physical parameters to improve design specifications given in terms of
    1. frequency of oscillation
    2. magnitude of oscillations
    3. magnitudes of various forces in the system including the forces in other elements, on walls, the total force on the mass.
19. Explain in both mathematical and physical terms the phenomenon or resonance
20. Explain force and displacement transmissibility concepts.
21. Identify, explain and compute transient and steady state solutions.
22. Use graphs from the book or supplied on the exam such as magnification factors, force and displacement transmissibilities and phase angles to:
    1. determine solutions to differential equations
    2. determine the change in the solution when a parameter changes.
23. Use a plot of a solution to a differential equation to
    1. determine the differential equation of which it is a solution
    2. identify parameters such as natural frequency and damping ratio.
24. Use a power series solution method to determine a solution to a differential equation.
25. Determine the radius of convergence for a series solution to a differential equation.
26. Use forms of solutions from the book to plug and chug answers (boring).
27. Solve the one-dimensional wave and heat condition equations with homogenous and inhomogeneous boundary conditions and sketch the solutions.
28. Identify and explain how parameter changes in the wave and heat condition equations will affect the solutions including effects on
    1. frequency of oscillation
    2. rate of decay
29. Use the method of separation of variables to solve partial differential equations.
30. Write a computer program to solve a single of multiple first order differential equations including
    1. Euler's method
    2. Taylor series methods
    3. Runge-Kutta methods.
31. Explain how the step size affects the error for each of the numerical methods listed above.