MAT 221B Midterm: Math 22B Midterm 1 Spring 2017
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Sample problems: midterm 1: for each of the following odes for y(x), say if it is linear or nonlinear and give its order. For what x-interval does it exist? (b) solve the initial-value problem y = xy3, y(0) = 1. For what x-interval does it exist: suppose that y(x) satis es the ode (a) show that the substitution y = y2. X. (1) y(x) = w (x) w(x) 1 gives the following ode for w(x) w . Xw = 0. (2) (b) classify the odes in (1) and (2): (a) water evaporates from a spherical raindrop at a rate proportional to the surface area of the raindrop. Write a di erential equation for the volume of the raindrop as a function of time. (a sphere of radius r has surface area. A = 4 r2 and volume v = 4 r3/3. ) (b) show that, according to this model, the raindrop will evaporate com- pletely after a nite time.
