Here is part of an approximate direction field for the first order differential equation \((4y^3-x)y^{\,\prime }=y-2x\).

A direction field.

Figure 1: A direction field
(a)
Let’s draw the integral curve in the direction field passing through \((-1,0)\).
(b)
Is this differential equation linear? Separable? Bernoulli? Nonlinear homogeneous?
(c)
Let’s rewrite this equation in the form \(M\,dx+N\,dy=0\).
(d)
Let’s check for exactness.
(e)
Let’s solve this equation for an implicit general solution.
(f)
Suppose \(y(x)\) satisfies \((4y^3-x)y^{\,\prime }=y-2x\), \(y(-1)=0\). Let’s find the solution(s) to \(y(x)=1\).