AME 301 Course Objectives

1. Identify a given differential equation as
   1. linear or nonlinear;
   2. homogeneous or inhomogeneous;
   3. variable or constant coefficient; and,
   4. ordinary or partial.
2. Classify a differential equation according to the solution techniques applicable to it, including
   1. seeking exponential solutions of the form exp(r t);
   2. using the method of undetermined coefficients;
   3. using the method of variation of parameters;
   4. using the Laplace transform method;
   5. using a numerical approximation approach; and
   6. using a method (perhaps unknown to you) that has not been considered in Chapters 1, 2, 3, 6 or 8 of the course text.
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;
   4. predict directly from the coefficients of the characteristic equation whether or not the zero solution to the homogeneous equation is asymptotically stable or unstable; and,
   5. explain why a non-zero Wronskian indicates that two solutions form a fundamental set of solutions.
4. Linearize a second order nonlinear differential equation about an equilibrium point, including
   1. identifying whether or not a point is an equilibrium point;
   2. explain the relationship between the nonlinear equation and the best linear approximation, including the limitations and benefits of the best linear approximation;
   3. use a Taylor series expansion of the nonlinear terms to determine the best linear approximation; and,
   4. use a Jacobian calculation to determine the best linear approximation.
5. For second order linear ordinary differential equations, apply Theorem 3.2.1 from the course text to
   1. identify when solutions are guaranteed to exist;
   2. identify when solutions are unique; and,
   3. determine the time interval upon when unique solutions are guaranteed to exist.
6. Explain and derive Euler's formula for exponentials of complex numbers.
7. From sufficiently complete plot of the response of a system, identify whether or not an equilibrium solation is stable, asymptotically stable or unstable.
8. Explain the relationship between the stability of an equilibrium solution of a nonlinear system and the best linear approximation about that equilibrium point.
9. For a second order, linear, constant coefficient homogeneous ordinary differential equation, determine the stability of the zero solution directly by inspection of the differential equation.
10. 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. identify when the method of variation of parameters is an appropriate solution technique;
    6. explain why the method of variation of parameters works for the class of differential equations to which it is applicable;
    7. use the method of variation of parameters to find a particular solution;
    8. use the method of variation of parameters to find the solution to an initial value problem;
11. Compute Laplace transforms of functions using the definition of the Laplace transform.
12. Determine the solution to the initial value problem for linear, constant coefficient differential equations including:
    1. homogeneous equations;
    2. inhomogeneous equations with
       1. continuous inhomogeneous terms;
       2. discontinuous inhomogeneous terms; and,
       3. impulsive inhomogeneous terms.
13. Solve a linear ordinary differential equation using Laplace transform technique, including, if necessary
    1. computing and/or using a table to determine the Laplace transform of a time function;
    2. applying the properties of Laplace transforms to convert a differential equation into an algebraic equation in the frequency domain;
    3. compute the partial fraction expansion of a function in the frequency domain; and,
    4. computing and/or using a table to determine the inverse Laplace transform of a frequency domain function.
14. Identify transfer function, Y(s) to which the Final Value theorem can be used to compute y(t) as time becomes large.
15. Apply the Final Value theorem to determine the steady state response of a transfer function, Y(s)/R(s) to different reference inputs, R(s) (steps, ramps, etc.).
16. Express the solution of an initial value problem from linear, constant coefficient differential equations in terms of a convolution integral.
17. Write a computer program to solve an initial value problem for linear, first order differential equations including
    1. using Euler's method;
    2. using improved Euler's method;
    3. using the fourth order Runge-Kutta method;
    4. using the above methods to solve systems of first order equations; and,
    5. converting second order (and higher order) differential equations into a system of first order equations.
18. Derive a formula for the the local truncation error for the numerical methods covered in class.
19. Derive the equation(s) of motion for mechanical, electrical and electro-mechanical systems using
    1. Newton's laws;
    2. Kirchoff's current and voltage laws;
    3. Rayleigh's energy method;
    4. D.C. motor laws; and,
    5. tables and/or calculations to determine the equivalent spring constant and equivalent mass for various arrangements of elastic elements and inertial elements.
20. Determine the free vibration response of a single degree of freedom mechanical systems, including
    1. translational systems; and,
    2. torsional systems.
21. Determine the free vibration response of a single degree of freedom electrical system.
22. Determine the harmonically forced response of a single degree of freedom mechanical or electrical system.
23. Determine the natural frequency, damped natural frequency and damping ratio from a given second order differential equation.
24. Determine the natural frequency, damped natural frequency and damping ratio from the pole locations of a given second order differential equation.
25. Determine the approximate response of a mechanical system subjected to Coulomb damping.
26. Derive and solve the equations of motion for a mechanical system subjected to harmonic base motion.
27. For a mechanical system subjected to harmonic base motion, compute the force and displacement transmitted from the base motion to the system.
28. Compute the transfer function between a specified input and output for a mechanical, electrical or electro-mechanical system.
29. Determine the qualitative nature of the impulse and step response of a second order system directly from the pole locations of its corresponding transfer function.
30. List and explain the cause of the effects of varying the individual gains in a PID controller, on
    1. rise time;
    2. maximum overshoot;
    3. settling time; and,
    4. steady state tracking.
31. Determine the transient response characteristics (to a step input) of a transfer function based on a plot of the location of the transfer function's poles and zeros.
32. Given a transfer function, determine the response of the system to a specified input.
33. Determine the transfer function from the input to the output of a block diagram.