Sample Test 1 for General Physics Laboratory I | PHYS 221, Exams of Physics

Download Sample Test 1 for General Physics Laboratory I | PHYS 221 and more Exams Physics in PDF only on Docsity! Mathematics 221 Fall 04 Sample Test 1 1. For which value(s) of the constant b does the DE y0 + 3y = e2t have a solution of the form y = be2t? 2. For the DE in Problem 1 what is the slope of the minitangent in the slope …eld at the point (0; 5) in the ty plane? 3. Let the function y = y(t) be de…ned by the integral y(t) = Z t 1 1p s+ 1 sin 2sds: Find the derivative y0(t) of the function y(t). 4. Find the solution of the IVP y0 = 5ay; y(0) = y0, where a and y0 are constants. 5. Use separation of variables to solve y0 = y2 cost; y(0) = 1. 6. Find the general solution of the DE y0 = (t+y)2 by making a substitution v = t+ y. 7. For the DE y0 = y2 et draw the zero isocline (the set of points where the slope …eld is zero) in the ty plane. 8. What is the interval of existence of the solution to the IVP (t 3)3y0 t(t 2)y = cos t; y(1) = 0? 9. The solution curves to a certain DE are y = t2+C, where C is an arbitrary constant. Sketch a rough plot of the slope …eld of the DE in the window 2 < t < 2; 4 < y < 4: 10. Consider a “population" model dp dt = (p 0:5)(p 12)2: (a) What are the equilibrium populations? (b) Draw the phase line and classify any equilibria at asymptotically stable, unstable, or semistable. (c) Sketch a graph of the per capita growth rate 1p dp dt as a function of the population p. (d) Draw a rough time series plot (p versus t) of the solution p = p(t) if p(0) = 2. In reality, if this were a real population, what would you expect the population to be after a long time? (e) For which initial conditions p(0) does the population become extinct? 1