MATH 122
TEST III 

1.(30)  Solve the following differential equations.

a)  [(dy)/( dx)] = 2(xy-10x)

b)   y¢¢-10y¢+25y = 0  ,      y(0) = 2 ,    y¢(0) = 3

c)  y¢¢+3y¢+3y = 0

2.(10)  The equation in question #1b cannot be the differential equation for a "spring-mass" system.  Why not?

3.(20)  Let's model an ecosystem with two species.  Let x(t) be the population of species X at time t and y(t) the population of species Y.  Assume that in the absence of Y, species X shrinks exponentially (i.e., the 'growth' rate is proportional to the negative of the size of the population); assume that in the absence of X, species Y shrinks logistically.  However the two species actually help each other to survive in that each is positively affected by the presence of the other species.  For each species, this part of the growth rate is proportional to the product of the sizes of the populations.

a)  Write two differential equations that can be used to describe this system.

[(dx)/( dt)] =                  [(dy)/( dt)] =

b)  Your model contains (or it should contain) several constants.  Data would be needed to find the constants explicitly.  But even without data, tell the signs of the constants in the dx/dt equation and give reasons for the sign.

4.(10)  Consider the differential equation

dy
dx		 = x+y2.
 
a)  Use Euler's method with two steps to approximate the value of y when x = 3 on the solution curve that passes through (2,3).  Explain clearly what you are doing on a sketch.  Your sketch should show the coordinates of all the points you have found.

b)  Are your approximate values of y an under- or over-estimate?  Explain how you know this.
 

5.(20)  Consider the system:   [(dx)/( dt)] = 2x(2-x-y)    and    [(dy)/( dt)] = 3y(4-2x-y)

a)  Find all equilibria and nullclines.

b)  Sketch the nullclines.  (Note different scales on axes.)

c)  Sketch two trajectories:  one starting at (2,3) and the other starting at (1/2,1/2).
 

6.(10)  Consider the two differential equations: (A) [(dy)/( dx)] = y(2-y)  and

(B)  [(dy)/( dx)] = y(y-2).  Use the slope program and find the slope field for each of these differential equations.  (You do not have to show this.)

a)  Sketch a few solutions to each differential equation.

b)  The equations have obvious algebraic similarities.  Explain both the similarities and differences in the solutions by relating the pictures and the differential equations.

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