The University of Massachusetts Amherst
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Exams

Final exam

When: Monday 10 May 10:30 a.m.–12:30 p.m.

Where: Hasbrouck 126

Bring: pens, pencils, calculator (+ extra batteries), and your UMass ID card.

Help:

The Math 331 TAs will continue to hold their regular 5:00–7:00 p.m. office hours this week Monday–Thursday, May 3–6. See the About page for locations the different days.

What:

Additional sample questions for final exam. Correction: In each of #7 (a) and b), there should be a factor of y after the second (t+2).

You should also look again at the sample questions for the mid-semester exam (see below). It will not be possible for me to prepare solutions of these additional questions!

The entire semester’s work, including everything covered in the mid-semester exam.

Additional lectures covered: through end of semester

Additional homework covered: Homework sets 10–2 .

Additional textbook coverage:

  • Sections 3.5–3.8.
  • Sections 6.1–6.6.
  • Sections 7.1  and 7.4–7.8; section 9.1; and use of nullclines as in Sections 9.4–9.5.

Additional key things you should know how to do:

  • given one solution of a 2nd-order homogeneous linear ODE, use reduction of order to find another solution which, together with the given one, forms a fundamental set of solutions;
  • use the characteristic equation to solve constant-coefficient homogeneous linear 2nd-order ODEs (and IVPs) when there is a pair of conjugate complex characteristic roots, and to use those to obtain the general solutions in terms of real-valued functions;
  • find the general solution of a nonhomogeneous linear 2nd-order ODE given the general solution of its associated homogeneous ODE and a particular solution of the given nonhomogeneous ODE;
  • use linearity to explain why the preceding key thing works;
  • use the method of undetermined coefficients to find a particular solution of a non-homogeneous linear 2nd-order ODE having constant coefficients and one of the special-form forcing functions;
  • given a fundamental set of solutions the associated homogeneous ODE, use the method of variation of parameters to find a particular solution of a non-homogeneous linear 2nd-order ODE (where the method of undetermined coefficients does not apply);
  • model mechanical and electrical vibrations by means of 2nd-order IVPs;
  • analyze the behavior of solutions of such models (e.g., period, amplitude, and phase of a simple harmonic oscillator; types of damping; transient and steady-state components of solutions for a forced system; resonance and beats;
  • calculate the Laplace transform of a function by using the definition (as an improper integral);
  • use linearity and other basic properties of Laplace transforms together with a table of Laplace transforms to find the Laplace transform of a function;
  • apply Laplace transforms to solve 1st-order and 2nd-order linear IVPs;
  • represent discontinuous functions in terms of unit step functions;
  • represent impulse functions by the Dirac delta, and use Laplace transforms to solve linear IVPs involving impulse functions;
  • find the inverse Laplace transform of a product by using convolution;
  • given a single 2nd-order ODE, write an equivalent system of two 1st-order ODEs;
  • write a system of two 1st-order linear ODEs in equivalent matrix-vector form;
  • show that, for scalar ? and vector v, the function  e?t v is a solution of a homogeneous, liner, constant-coefficient vector ODE Y‘ = A Y if and only if ? is an eigenvalue of the coefficient matrix A and V is an associated eigenvector;
  • find eigenvalues and eigenvectors for a 2×2 matrix A and thereby find straight-line solutions Y = of  e?t V of Y‘ = A Y;
  • find a fundamental set of solutions and the general solution of Y‘ = A Y by using the eigenvalues and eigenvectors of A;
  • solve an initial-value problem Y‘ = A Y, Y(0) = v0;
  • given eigenvalues and eigenvectors for a 2×2 matrix A, sketch and describe the phase portrait of  Y‘ = A Y;
  • find the nullclines for a system of two linear or non-linear 1st-order ODEs;
  • use nullclines for such a system to describe qualitatively the behavior of trajectories through points in the various regions into which the nullclines separate the plane.

Final exam conflicts

The deadline to notify me of any final exam conflict Monday 26 May. This is in accordance with published course syllabus (printed and the About page here):

“Obtain from the Registrar’s Office a certified copy of your entire final exam schedule that indicates the conflict. On that form write your e-mail and phone contact information as well as the name of the instructor in the course creating the final conflict. Give this form to me no later than two weeks before the Math 331 final exam.”

Mid-semester exam

Get mid-semester exam solutions

To be given during class Wednesday, March 10 (snow date Monday, March 22).

Sample questions for mid-semester exam

Get solutions to sample questions

Lectures covered: through Wednesday 3 March

Homework covered: Homework sets 1–9. (Change!)

Textbook coverage:

  • Sections 1.1–1.3
  • Sections 2.1–2.5
  • Section 2.6 but not the subsection at end on using integrating factors to make ODEs exact
  • Sections 2.7 and (lightly) 8.1–8.3: you do need to know the formula for Euler’s Method but not the formulas for the Improved Euler or RK4 methods
  • Sections 3.1–3.4

Key things you should know how to do:

  • check that an alleged solution of an ODE or IVP really is such;
  • distinguish between the set of all solutions (“the general solution”) of an ODE and the particular solution satisfying an IC (or ICs);
  • for a 1st-order ODE, find the slope of the solution curve through a point as the value of the right-hand side of the ODE at that point;
  • construct a slope field for a 1st-order ODE and use it to sketch typical solutions and describe the behavior of these solutions (e.g., what happens as t ? ?);
  • find and classify equilibrium solutions for a 1st-order ODE;
  • use separation of variables to solve a separable 1st-order ODE;
  • use an integrating factor to solve a linear 1st-order ODE;
  • determine whether a 1st-order ODE is exact and, if so, find its solution; (topic omitted)
  • use Euler’s method to find approximate solutions of a 1st-order IVP;
  • distinguish between local and global truncation error for a numerical method;
  • tell what it means to say that such an error is O(hk) and what happens if the step-size h is changed by a given factor;
  • construct simple 1st-order ODE models for simple real-world phenomena (e.g., Newton’s Law of Cooling, mixing problems, population models); solve the ODE; and analyze the behavior of solutions;
  • know that any linear combination y = c1 y1 + c2 y2 of solutions y1 and y2 of a homogeneous linear 2nd-order ODE is also a solution—and why;
  • calculate the Wronskian W[y1, y2] of two functions y1 and y2 and know that, when they are solutions of the same homogeneous linear 2nd-order ODE, then the Wronskian is either nowhere 0 or else everywhere 0 (on the relevant interval);
  • know how to obtain the general solution of a homogeneous linear 2nd-order ODE when you have two particular solutions y1 and y2 whose Wronskian is nonzero; and
  • use the characteristic equation to solve constant-coefficient homogeneous linear 2nd-order ODEs (and IVPs); and
  • do basic arithmetic of complex numbers and use Euler’s formula to express a complex exponential in terms of cosine and sine.