- Identify a given differential equation as
- linear or nonlinear;
- homogeneous or inhomogeneous;
- variable or constant coefficient; and,
- ordinary or partial.

- Classify a differential equation according to the
solution techniques applicable to it, including
- seeking exponential solutions of the form exp(r t);
- using the method of undetermined coefficients;
- using the method of variation of parameters;
- using the Laplace transform method;
- using a numerical approximation approach; and
- using a method (perhaps unknown to you) that has not been considered in Chapters 1, 2, 3, 6 or 8 of the course text.

- For linear, homogeneous constant-coefficient ordinary
differential equations
- explain and prove the principle of superposition;
- determine the general solution and solve an initial value problem for real, complex conjugate and repeated roots of the characteristic equation; and,
- 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; - 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,
- explain why a non-zero Wronskian indicates that two solutions form a fundamental set of solutions.

- Linearize a second order nonlinear differential equation
about an equilibrium point, including
- identifying whether or not a point is an equilibrium point;
- explain the relationship between the nonlinear equation and the best linear approximation, including the limitations and benefits of the best linear approximation;
- use a Taylor series expansion of the nonlinear terms to determine the best linear approximation; and,
- use a Jacobian calculation to determine the best linear approximation.

- For second order linear ordinary differential equations,
apply Theorem 3.2.1 from the course text to
- identify when solutions are guaranteed to exist;
- identify when solutions are unique; and,
- determine the time interval upon when unique solutions are guaranteed to exist.

- Explain and derive Euler's formula for exponentials of complex numbers.
- From sufficiently complete plot of the response of a system, identify whether or not an equilibrium solation is stable, asymptotically stable or unstable.
- Explain the relationship between the stability of an equilibrium solution of a nonlinear system and the best linear approximation about that equilibrium point.
- 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.
- For second order, linear constant coefficient inhomogeneous
equations
- identify when the method of undetermined coefficients is an appropriate solution technique;
- explain why the method of undetermined coefficients works for the class of differential equations to which it is applicable;
- use the method of undetermined coefficients to find a particular solution;
- use the method of undetermined coefficients to find the solution to an initial value problem;
- identify when the method of variation of parameters is an appropriate solution technique;
- explain why the method of variation of parameters works for the class of differential equations to which it is applicable;
- use the method of variation of parameters to find a particular solution;
- use the method of variation of parameters to find the solution to an initial value problem;

- Compute Laplace transforms of functions using the definition of the Laplace transform.
- Determine the solution to the initial value problem for
linear, constant coefficient differential equations including:
- homogeneous equations;
- inhomogeneous equations with
- continuous inhomogeneous terms;
- discontinuous inhomogeneous terms; and,
- impulsive inhomogeneous terms.

- Solve a linear ordinary differential equation using Laplace
transform technique, including, if necessary
- computing and/or using a table to determine the Laplace transform of a time function;
- applying the properties of Laplace transforms to convert a differential equation into an algebraic equation in the frequency domain;
- compute the partial fraction expansion of a function in the frequency domain; and,
- computing and/or using a table to determine the inverse Laplace transform of a frequency domain function.

- Identify transfer function,
*Y(s)*to which the Final Value theorem can be used to compute*y(t)*as time becomes large. - 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.*). - Express the solution of an initial value problem from linear, constant coefficient differential equations in terms of a convolution integral.
- Write a computer program to solve an initial value
problem for linear, first order differential equations
including
- using Euler's method;
- using improved Euler's method;
- using the fourth order Runge-Kutta method;
- using the above methods to solve systems of first order equations; and,
- converting second order (and higher order) differential equations into a system of first order equations.

- Derive a formula for the the local truncation error for the numerical methods covered in class.
- Derive the equation(s) of motion for mechanical, electrical
and electro-mechanical systems using
- Newton's laws;
- Kirchoff's current and voltage laws;
- Rayleigh's energy method;
- D.C. motor laws; and,
- tables and/or calculations to determine the equivalent spring constant and equivalent mass for various arrangements of elastic elements and inertial elements.

- Determine the free vibration response of a single degree of
freedom mechanical systems, including
- translational systems; and,
- torsional systems.

- Determine the free vibration response of a single degree of freedom electrical system.
- Determine the harmonically forced response of a single degree of freedom mechanical or electrical system.
- Determine the natural frequency, damped natural frequency and damping ratio from a given second order differential equation.
- Determine the natural frequency, damped natural frequency and damping ratio from the pole locations of a given second order differential equation.
- Determine the approximate response of a mechanical system subjected to Coulomb damping.
- Derive and solve the equations of motion for a mechanical system subjected to harmonic base motion.
- For a mechanical system subjected to harmonic base motion, compute the force and displacement transmitted from the base motion to the system.
- Compute the transfer function between a specified input and output for a mechanical, electrical or electro-mechanical system.
- 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.
- List and explain the cause of the effects of varying the
individual gains in a PID controller, on
- rise time;
- maximum overshoot;
- settling time; and,
- steady state tracking.

- 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.
- Given a transfer function, determine the response of the system to a specified input.
- Determine the transfer function from the input to the output of a block diagram.

B. Goodwine J. Lucey Last modified: Sun Dec 14 11:15:23 EST 2003