Let’s consider the first order Bernoulli differential equation \(xy^{\,\prime }=y-2xy^2\).

(a)
Does this equation have any constant solutions?
(b)
Let’s look for non-constant solutions. Let’s write this equation the standard form \(y^{\,\prime }+p(x)y=q(x)y^r\) and determine a non-trivial solution \(y_1\) to the complementary equation \(y^{\,\prime }+p(x)y=0\).
(c)
Let’s use the substitution \(u=\dfrac {y}{y_1}\) to determine a separable differential equation \(u\) must satisfy.
(d)
Let’s solve that equation, and then determine a general solution for \(y\).
(e)
What is the solution to the IVP \(xy^{\,\prime }=y-2xy^2\), \(y(1)=0.5\)?
(f)
What is the solution to the IVP \(xy^{\,\prime }=y-2xy^2\), \(y(1)=0\)?