Various DE problems and related stuff - help

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My DE class has been 100% about solving equations giving an equation. But my teacher seems to think giving us a very wordy sheet of homework would be fun and I just seem to be very lost. Trying to go through this systematically starting with the first problem and I've no idea what to do.

I have to draw a direction field. I figured I could plug in various values of x and t to just get dx/dt, but I don't know what to do about the other variables, r, b, and a. Can I simply set r=1, a=1, and b=0, arbitrarily, since they would be random constants in the logistics equation in the first place? So lost.

Attached is all the assignment.
 

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Yes, a, b, and r are constants. I would be a little careful about setting any constant to 0. That may remove important parts of the equation and so not be as general as you would like to.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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