how to transform 2xdy - y(x+1)dx + 6y^3dx = 0 into a Bernoulli Equation??

A bernoulli equation has the general form:

y' + p(x)y = q(x)yn

Examine the given equation to see what needs to be done to do the transformation, and then identify what is p(x) and q(x).

To transform the given differential equation, 2xdy - y(x+1)dx + 6y^3dx = 0, into a Bernoulli Equation, you need to find a suitable substitution that allows you to rewrite the equation in a specific form.

A Bernoulli Equation is of the form dy/dx + P(x)y = Q(x)y^n, where n is a constant.

Let's follow these steps to transform the equation into the Bernoulli form:

Step 1: Divide the entire equation by dx to separate the differentials. This gives us:
2xdy/dx - y(x+1) + 6y^3 = 0

Step 2: Rearrange the terms by moving the y(x+1) term to the left-hand side and the 2xdy/dx term to the right-hand side:
-y(x+1) = -2xdy/dx - 6y^3

Step 3: Divide the equation by y^3 to isolate the dy/dx term:
(-1/y^3)(x+1) = (-2/x)(dy/dx) - 6

Step 4: Make a substitution using v = y^(-2):
Differentiate both sides of the equation with respect to x:
dv/dx = -2y^(-3)(dy/dx)

Step 5: Substitute this into the previous equation:
(-1/(v^2))(x+1) = (-2/x)(dv/dx) - 6

Step 6: Simplify the equation and rearrange it to match the Bernoulli Equation form:
-2(x+1)/xv^2 + 6 = dv/dx

Now, you have transformed the given equation into the form of Bernoulli Equation by using the substitution v = y^(-2).