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Saturday, 3 September 2011

ODE - Assignment 1

Due Date: September 9 September 12
  1. A tank contains 800 gal of water in which 200 lb of salt is dissolved. Two gallons of fresh water runs in per minute and 2 gal of the mixture in the tank, kept uniform by stirring, runs out per minute.
    1. Form a differential equation for the amount of salt in the tank as a function of time.
    2. How much salt is left in the tank after 5 hours?
  2. A thermometer, reading 10^\circ\mathrm{C}, is brought into a room whose temperature is 23^\circ\mathrm{C}. Two minutes later the thermometer reading is 18^\circ\mathrm{C}.
    1. Use Newton's Law of Cooling to model the change of the thermometer reading with time.
    2. How long will it take until the reading is practically 23^\circ\mathrm{C}, say, 22.8^\circ\mathrm{C}?
  3. Solve the following initial value problem (IVP) and make a reasonably accurate sketch of the solution:
    2y^\prime + y^3 = 0;\quad y(0)=1
  4. Solve the following ODEs:
    1. \left[\sin(y)\cos(y)+x\cos^2(y)\right]\,dx + x\,dy=0
    2. e^y\,\left[\sinh(x)\,dx + \cosh(x)\,dy\right]=0
    3. (x^2+y^2)\,dx - 2xy\,dy =0
    4. \left(\cos(xy)+\dfrac{x}{y}\right)\,dx + \left(1 + \dfrac{x}{y}\cos(xy)\right)\,dy=0
    1. Under what conditions for the constants A,B,C,D, is the following ODE exact?
      (Ax+By)\,dx + (Cx+Dy)\,dy=0
    2. Solve this exact ODE.
    1. Solve the ordinary differential equation: y^\prime\tan(x)=2y-8.
    2. Sketch the family of solutions given by the general solution to the above ODE.
    3. Give the particular solution to the above ODE such that y=0 when x=\pi/2.

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