Objectives for Final Exam

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To successfully complete the final exam in AME 30314, you should be able to:

1. Identify a given differential equation as
   1. linear or nonlinear;
   2. homogeneous or inhomogeneous;
   3. variable or constant coefficient;
   4. ordinary or partial; and,
   identify its order.
2. Classify a differential equation according to the solution techniques applicable to it, including
   1. using integrating factors;
   2. using the fact that it is separable;
   3. using the fact that it is exact;
   4. converting an equation that is not exact into one that is exact if given an appropriate integrating factor;
   5. seeking exponential solutions of the form exp(r t);
   6. using the method of undetermined coefficients;
   7. using the method of variation of parameters;
   8. using a numerical approximation approach; and,
   9. using the separation of variables method for partial differential equations
   and use these methods to determine general solutions and solve initial value problems.
3. For linear, homogeneous constant-coefficient ordinary differential equations
   1. explain and prove the principle of superposition;
   2. determine the general solution and solve an initial value problem for real, complex conjugate and repeated roots of the characteristic equation; and,
   3. identify directly from the roots of the characteristic equation the qualitative nature of either the general solution or the solution to the initial value problem, i.e., whether or not the solution will be oscillatory and whether or not the solution will grow or decay in time; and,
   4. explain why a non-zero Wronskian indicates that two solutions form a fundamental set of solutions.
4. Linearize a second order nonlinear differential equation by computing a Taylor series expansion of the nonlinear terms to determine the best linear approximation.
5. Explain and derive Euler's formula for exponentials of complex numbers.
6. For second order, linear constant coefficient inhomogeneous equations
   1. identify when the method of undetermined coefficients is an appropriate solution technique;
   2. explain why the method of undetermined coefficients works for the class of differential equations to which it is applicable;
   3. use the method of undetermined coefficients to find a particular solution;
   4. use the method of undetermined coefficients to find the solution to an initial value problem;
   5. use the method of undetermined coefficients to find a general solution;
   6. identify when the method of variation of parameters is an appropriate solution technique;
   7. explain why the method of variation of parameters works for the class of differential equations to which it is applicable;
   8. use the method of variation of parameters to find a particular solution; and,
   9. use the method of variation of parameters to find the solution to an initial value problem.
7. Write a computer program to solve an initial value problem for linear differential equations including
   1. using Euler's method;
   2. using Taylor series methods;
   3. using the fourth order Runge-Kutta method;
   4. using the above methods to determine approximate solutions to a first order equation;
   5. using the above methods to determine approximate solutions for systems of first order equations; and,
   6. converting second order (and higher order) differential equations into a system of first order equations.
8. Derive a formula for the the local truncation error for the numerical methods covered in class.
9. Explain and identify the effect of the step size on the error in an approximate numerical solution for the above numerical methods.
10. Explain and identify the difference between local truncation error and the global error for the above numerical methods.
11. Determine the free vibration response of a single degree of freedom mechanical systems.
12. Determine the harmonically forced response of a single degree of freedom mechanical or electrical system.
13. Determine the natural frequency, damped natural frequency and damping ratio from a given second order differential equation.
14. Derive and solve the equations of motion for a mechanical system subjected to harmonic base motion.
15. From a plot of the damped oscillatory response of a second order system, identify the damped natural frequency, the damping ratio and the natural frequency.
16. For a mechanical system subjected to harmonic base motion, compute the force and displacement transmitted from the base motion to the system.
17. For a given system with a specified input, write an expression for a PID controller for the input.
18. For a PID controlled system, determine gain values that will result in an oscillatory or a non-oscillatory response.
19. For a PID controlled system, determine gain values that will result in a specified steady state error for the system.
20. For a PID controlled system, determine gain values that will result in a specified damping ratio for the controlled system.
21. Solve partial differential equations using the separation of variables method including
    1. the wave equation;
    2. the heat conduction equation; and,
    3. Laplace's equation,
    with various combinations of boundary and initial conditions and in one and multiple dimensions.
22. For a given periodic function, compute a Fourier series representation for that function.
23. Identify when a given periodic funcation will have only sine or cosine terms in a Fourier series representation for it.