Mathematics

Material on Test 2

Test 2 is Thursday, Nov. 2. The test will be on Secs. 4.1 - 4.6, 4.8 - 4.9, 7.1 - 7.6, 7.8

Calculators may be used for arithmetic only.

Bring your text to the test. You may use the table of integrals in the front of the text, and the table of Laplace transforms in the back of the text.

The properties of Laplace transforms that we use are all in the table in the back of the text:
• Theorem 3 on p. 360 is formula 2 in the table.
• Theorem 4 on p. 361 is formula 3 in the table.
• Theorem 5 on p. 362 is formula 4 in the table.
• Theorem 6 on p. 363 is formula 5 in the table.
• Formulas (5) and (8) on p. 387 are formulas 10 and 11 in the table.

A brief description of the method of undetermined coefficients is on the inside back cover of the text; you are welcome to refer to it, but it is very brief!

A version of variation of parameters is also described on the inside back cover of the text. Since it is not the version we used, I will provide you with the variation of parameters formula we used. It is in the online "Notes on Delta Functions" at the top of p. 10.

Things you should be able to do:

Secs. 4.2 - 4.3. Find the general solution of ay''+by'+cy=0. Determine the values of the arbitrary constants needed to solve an initial value problem.

Sec. 4.4. Use the method of undetermined coefficients to find a particular solution to a 2nd order equation.

Sec. 4.5. Use the method of undetermined coefficients and the superposition principle to find the general solution to a 2nd order equation, and use the general solution to solve initial value problems.

Sec. 4.6. Use the variation of parameters formula that was presented in lecture (not the version in the text) and the superposition principle to find the general solution to a 2nd order equation.

Secs. 4.8 - 4.9. Solve initial value problems for spring-mass systems, and interpret the answer.

Sec. 7.2. Calculate Laplace transforms from the definition and from the table.

Sec. 7.3. Calculate Laplace transforms of e^(at) f(t), derivatives of f(t), and t^n f(t).

Sec. 7.4. Calculate inverse Laplace transforms using the table and partial fractions.

Sec. 7.5. Use Laplace transforms to solve initial value problems for 2nd order equations.

Sec. 7.6. Calculate Laplace transforms of discontinuous functions. (Omit periodic functions.) Use Laplace transforms to solve initial value problems for 2nd order equations when the forcing function is discontinuous.

Sec. 7.8. Use Laplace transforms to solve initial value problems for 2nd order equations when the forcing function is a delta function.

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