$\"\"$

(a)

The function $\"\"$ is proportional to the product of $\"\"$ and the difference between $\"\"$ and $\"\"$ .

The function is $\"\"$ .

Rewrite the function.

$\"\"$

Apply integral on each side.

$\"\"$

Integrate power rule: $\"\"$ .

$\"\"$

(b)

The direction field satisfies $\"\"$ along $\"\"$ and $\"\"$ .

The differential equation matches with the slope field $\"\"$ .

$\"\"$

(c)

Verification :

(1) Draw the coordinate plane.

(2) Graph the differential equation $\"\"$ .

Graph :

$\"\"$

Observe the graph :

The differential equation matches with the slope field $\"\"$ .

$\"\"$

(a) $\"\"$ .

(b) The differential equation matches with the slope field $\"\"$ .