1. Solve the intitial value problem: -4x^2 + y^2/xy, y(1) = -3 2. A 60 gallon ta

Question

1. Solve the intitial value problem: -4x^2 + y^2/xy, y(1) = -3

2. A 60 gallon tank contains 30 gallons of a solution that has 6 pound of salt dissolved in it. A solution that has a salt concentration of 0.3 pounds per gallon flows into the tank at a rate of 4 gallons per minute. The thoroughly mixed content of the trunk flows out at a rate of 3 gallons per minute. What is the amount of salt in pounds in the tank when the tank is full?

3. Given the differential equation perhaps modeling a population of money (population could be negative) give the equilibrium solutions. Which are stable, which are unstable? Sketch a direction field (Slope field) and put in integral curves for the various types initial conditions y(0) = yo. dy/dt = y(y-6)/(3-y).

4. Show that one of the following differential equations is exact and that one is not. Show that using the integrating factor of u(x) = 1/x that the one that was not becomes exact and then solve that one.

a) (y-2x^2) + (xlnx - 2xy +)dy/dx = 0 b) (3xcosy - 2) + (-3/2x^2siny - y)dy/dx = 0

5. Solve the Bernoulli equation dy/dt + (1/t)y = t^4y^3 , y(-1) = 1 explicitly for y(t) using the process that the substitution v = y^1-N transforms a differential equation of the form dy/dt + p(t)y = q(t)y^N into an equation of the form dv/dt + (1- N)p(t)V = (1-N)q(t) .

Explanation / Answer

1. {y/x-4 x^2, y(1) = -3}

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