Differential Equations: Initial Value Problem Question. Please help!!?

Question

How do I start answering this question? I'm not sure how these types of problems work, and my professor has such a thick accent I can't understand him.

Verify that psi(x) = 2/(1-ce^(x)), where c is an arbitrary constant, is a one-parameter family of solutions to

dy/dx = (y*(y-2))/2

Graph the solution curves.

I don't understand why y is in the second equation, but not the first. I have no idea how to even begin this problem. Thank you!

Answer 1

There is obviously a typo in the problem.  You are supposed to verify that the function:

y(x) = 2/(1-c*exp(x))

is a solution to the differential equation:

y'(x) = y*(y-2))/2

Note that this differential equation is a form of the "logistic equation".  If you are interested in knowing more about this type of equation, do a search using your favorite search engine for the term "logistic equation", and you'll get lots of hits.

To show that the given function is a solution, simply take the deriviative and plug that into the original equation and see if the equation is satisfied:

y = 2/(1-c*exp(x))

y' = 2*c*exp(x)/(1-c*exp(x))^2

Now, at first glance, the right hand side of this equation doesn't look much like the right hand side of the original differential equation, but note that using the definition of y(x) we can write:

2/(1-c*exp(x))^2 = (y^2)/2

so

y' = c*(y^2)exp(x)/2

Again going back to the definition of y, we can write expx) in terms of y:

y = 2/(1-c*exp(x))

1-c*exp(x) = 2/y

exp(x) = (1/c)*(1 - 2/y) = (1/c)*(y-2)/y

Plug this into the previous equation to get:

y' = c*(y^2)*(1/c)*(y-2)/(2*y)

y' = y*(y-2)/2

Viola! We have the original differential equation back, so this is a solution.

You are on your own for plotting the curves.  You should find that the solution is undefined for x = 0.  (The limit as x-> 0 is undefined -- the solution goes to +infinity if the limit is approached from the -x direction, and to -infinity is the limit is approached from the +x direction).   As x -> + infinity, y(x)-> 0, and as x -> -infinity, y(x) -> 2.