1) Write down a differential equation of the form y' = ay + b whose...

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1) Write down a differential equation of the form y' = ay + b whose solutions have the required behavior as t->oo, then, sketch a direction field and justify your answer.
a) All solutions y (t) converge to y = 1/2,b) all solutions diverge from y = 1/2 for y (t) ≠ 1/2
2) Solve the following initial value problems and sketch the graph of the solution on the same set of the coordinate axes and describe the behavior of the solutions as t->oo. You must show your work.
y' = -2y+8, a) y (0) = 1,   b) y(0) =4,   c) y(0) = 5
3) Solve the following initial value problems and sketch the graph of the solution on the same set of the coordinate axes and describe the behavior of the solutions as t->oo . You must show your work.
y' = 2y-8, - a) y (0) = 4, y (0) = 1, y (0) = 5
4) A tank initially contains 100 gal of water and 30 g of salt. Water containing a salt concentration of 6g/gal flows into the tank at the rate of 10 gal /h, and the well stirred mixture in the tank flows out at the same rate 10 gal/ h. You must show your work.
(a) Write a differential equation for the amount of salt in the tank at any time t.
(b) Solve the equation.
(c) Find the limiting amount of salt in the tank.

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