21-260 Differential Equations D. HandronExam #1 ReviewClosed book and notes; calculators not permitted. Be sure to show all work and explainyour reasoning as clearly as possible.1. Consider the initial value problemxy0+ 3y = x3, y(1) = 10(a) Find the general solution to the differential equation.(b) Find the particular solution to the initial value problem.2. (a) Use isoclines to draw the direction field for the differential equationy0=14x2+ y2− 1.Sketch the solution curve passing through the point (0, 0).(b) Sketch the direction field for the differential equation y0= (y − 1)(y + 1)(y + 2)2. Arethere any constant solutions? Why might you think so? How can you be certain?3. A system consists of three tanks containing salt solutions. The first tank holds 100` ofsolution, the second 100` and the third initially holds 50` of solution. Tank 1 initiallycontains 25g of salt in solution, Tank 2 contains 10g initially, and Tank 3 begins with 50g.Pure water is added to the first tank at a rate of 5` per minute. Two spigots allow thewater to flow from Vat 1 to Vats 2 and 3. The rate of flow for these spigots is 4` perminute and 1` per minute respectively. The solution flows from Vat 2 to Vat 3 at a rate of4` per minute. The solution from Vat 3 is allowed to flow onto the ground at a rate of 6`per minute (most likely destroying a fragile ecosystem, but that is none of our concern).Let xj(t) be the ammount of salt in tank j at time t.(a) Write three differential equations that describes the behovior of x1, x2and x3.(b) Verify that x1(t) = 25e−t/20.(c) Find the solution for x2(t).14. Consider the differential equationdydx= (y − 1)(y + 2)2(y2+ 1).(a) Draw the direction field for this differential equation.(b) Let y(t) be the solution satisfying the initial condition y(0) = 0. Can the value ofy(t) ever be less than −2? Why or why not?5. Consider the differential equationdydt= y + etThis is a linear differential equation. Find the solution satisfying the initial conditiony(1) =
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