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Berkeley MATH 1B - Separable Differential Equations

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Math 1B Worksheet 20:Separable Differential EquationsThursday, 6 November 2007GSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/Please introduce yourselves to each other, and put your names at the top of a piece ofblackboard. Take turns being the scribe: each of you should have a chance to write on thechalkboard for at least one of the exercises.These exercises are hard — harder than on the homework, quizzes, or exams. Thatmeans that you should spend some time thinking and talking about them; they’re designedto be solved in groups (the best way to learn mathematics). The problems are roughly inorder of increasing difficulty. I don’t expect any group to solve all of them.Don’t forget to draw pictures.1. Here are some differential equations to warm up with:(a) y0= (xny) / ((ln y)n)(b) y0= cos2(y/x) + y/x2. Consider the differential equationdydx= 2√yWhat is the general solution to this equation? What is the equation for y if y(3) = 1?By direct substitution, check that your equation really is a solution.What is the equation for y if y(3) = 0? Draw the direction field corresponding tothis differential equation.3. Last time, we discussed the production of ammoniaN2+ 3H2→ 2NH3When the concentration of ammonia is small (NH3(0) = 0), the rate of reaction iscontrolled by the concentrations of nitrogen and hydrogen:dNH3(t)dt= KN2(0) −12NH3(t)H2(0) −32NH3(t)31(a) If H2(0) = 3N2(0) (so that there is no limiting reactant), solve the differentialequation explicitly. In terms of K, how long does it take to convert half of thereactants to ammonia?(b) If H2(0) > 3N2(0) (so that ammonia is limiting), what’s the behavior of thesystem? What if hydrogen is limiting?4. Myoglobin, like hemoglobin, allows oxygen to dissolve in water. (The normal solu-bility is about 0.0001 mol O2per liter.) Hemoglobin transports oxygen through yourblood; myoglobin stores oxygen in muscle tissue.Mb(aq) + O2(g) ↔ Mb(O2)(aq)Write and solve a differential equation to describe the concentration of oxygenatedmyoglobin, assuming that this is the only reaction that occurs. Remember that thereaction can occur in both directions.5. The acceleration a = dv/dt on an object is equal to the total force F on the objectdivided by the object’s mass m. A raindrop falling through the air experiences aconstant downward force Fgravity= mg from gravity, and an upward force from airresistance proportional to the velocity Fresistance= kv. Terminal velocity occurs whenthe net acceleration is zero.(a) In terms of m, g, and k, solve for the terminal velocity.(b) Write and solve a differential equation describing the velocity of a raindrop thatstarts with zero velocity at time t = 0.(c) What if a particle starts with a downward velocity greater than terminal velocity,e.g. because it was thrown down?A marble falling through honey also has a terminal velocity, but in a viscous material,the resistance is proportional to the square of the velocity Fresistance= lv2. (Theeffective gravity is also reduced by buoyancy, but it is still a constant.) Repeat parts(a), (b), and (c) above in this setting.6. And, as always, the requisite turkey problem:A turkey, which starts at 70 degrees Fahrenheit, is placed in a 300-degree. The rate ofheat transfer between two materials is proportional to the difference in temperaturesof the materials (the proportionality constant depends on various parameters, likethe type of material and the size). The turkey is done when the internal temperaturereaches 180 degrees. If after an hour the internal temperature of the turkey is 86degrees, and dinner is at 5 p.m., when should you start cooking for


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