Spring 2006 - Terras' Class - Practice Exam 2

# Spring 2006 - Terras' Class - Practice Exam 2 - c Find the...

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aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa MATH 20D - Practice Exam #2 closed book, no calculators, no computers, no notes, . .. each problem is worth the same number of points aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 1) a) Find an integrating factor u for y’ - xy = -x. b) Solve the ODE in part a) by multiplying it by the integrating factor u and recognizing the left hand side as (uy)’. 2) Solve the separable ODE y y’ + x = 0 with initial condition y(2)=2. Find the interval a x b where the solution y(x) is defined. If you were to plot the direction field, what would you expect to see? 3) A tank contains 100 gallons of salty water made by dissolving 80 lb. of salt in water. Pure water runs into the tank at the rate of 4 gal./min, and the well-stirred mixture runs out at the same rate. Let Q(t) be the amount of salt in the tank at time t. a) Set up the ODE for this situation using Q’(t) = rate in - rate out. b) Find the formula for Q(t).

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Unformatted text preview: c) Find the time required for half the salt to leave the tank. Don’t compute the decimal approximation here. 4) True - False. Tell whether the following statements are true or false. Give a reason for your answer. a) The ODE (y cosx + 2x e y ) + (sinx +x 2 e y-1) y’(x) = 0 is exact. b) If f and f y ∂ ∂ are continuous in a rectangle |x-x | ≤ a, |y-y | ≤ b, then there is some interval on which the initial value problem y’(x) = f(x,y), f(x )=y , cannot have 2 distinct solutions. c) The logistic equation y’(x) = ry(1-y/K) has two equilibrium solutions y = K and y = 0. The first of these is stable while the second is unstable. 5) a) Find a fundamental set of solutions for y” + y’ - 2y = 0. b) Compute the Wronskian determinant of the fundamental set from part a). c) Solve the initial value problem: y” + y’ -2y = 0, y(0)=2, y’(0) = 3. aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa...
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Spring 2006 - Terras' Class - Practice Exam 2 - c Find the...

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