Please plan to attend the review section. More complete information
will be given there
Exam II covers sections 3.1 - 3.9 on second order linear
eqautions,
and sections 5.2 and 5.3 on series solutions near ordinary points.
Section 5.1 is a review of power series, and contains essential
material, though there are no direct questions on it.
Required skills for the test include:
Set up the characteristic eq'n for *) ay'' + by' = cy = 0 and
find its roots r1 and r2. Distinguish 3 cases.
Case r1 not = r2 real, solve equation and IVP(2 MC)
Know theory of solutions - FSS of *), linear independence of y1
and y2, Wronskian W(y1,y2) and relation of these concepts (1 MC)
Case r1, r2 are a conjugate pair of non- real roots. Find
solutions of eq'ns and IVPs (2 MC)
Case r1 = r2 = r real. Solve equations and IVPs (2 MC)
Considering non-hmogeneous Ly = g(t), distinguish eq'ns which
must be solved by variation of parameters, not undetermined coeffs.
(1 MC)
Solve non-homogen. eqn's using U. C. (2 MC)
Solve these using V.P. (1 MC)
Be generally familiar with the analysis of mechanical
oscillators, as outlined below (2 MC + HG)
Set up main equation *) mu'' + cu" + ku = F(t)
Solve *) in free undamped case, and find amplitude, freq., period and
phase shift
In damped free case distinguish the three levels of damping
In this case, put u(t) in an exponentially damped harmonic form,
and find quasi-freq. and quasi-period
Be familiar with the undamped forced case *) mu'' + ku = G(t)
where G(t) = F cos at or F sin at. Distinguish beat and resonant
solutions
In the case of damped forces motion with the above forcing
functions, find the steady state solution
Be familiar with power series solutions of *) P(x)y'' + Q(x)y' +
R(x)y = 0 expanded about x0 = 0, in the case 0 is an ordinary
point
Be able to adjust exponents in sums, collect terms and find a
recursion relation for the coefficients an of the series. (1 MC)
Find a lower bound on the radii of convergence of the series
solutions of a particular *) (1 MC)
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