General Solution for Differential Equation: 6x y^2 -6x + 3 y^2 -3

[intenzxboi]

Give the general solution of:

dy/dx = 6x y^2 -6x + 3 y^2 -3.

No clue how to start. Guessing you would use Bernoulli's equation but i don't see how it is possible to put in the y'(x) + p(x)y(x) = q(x) y(x)^n form.

 

[Gregg]

Bernoulli Equation

$$\frac{dy}{dx} = f(x)y+g(x)y^k$$

$$y^{1-k}=y_1 + y_2$$

where

$$\phi(x) = (1-k)\int f(x) dx$$

$$y_1 = Ce^\phi$$

$$y_2 = (1-k)e^\phi \int e^{-\phi} g(x) dx$$

 

[Gregg]

Actually don't bother, you can rearrange to get this:

$$\frac{dy}{dx} = (6x+3)(y^2-1)$$

you can take it from there, no?

 

[intenzxboi]

Yes i can thank you

 

[intenzxboi]

actually I am still having trouble do i still use the bernoulli method?

so it is a simple separation problem?

 

[Gregg]

Sorry I didn't realize you replied.

$$\frac{dy}{dx} = (6x+3)(y^2-1)$$


$$\int \frac{dy}{y^2-1} = \int (6x+3) dx$$


$$\frac{1}{2}log\left(\frac{1-y}{1+y}\right) = 3x^2+3x + C$$


$$\sqrt{\frac{1-y}{1+y}} = Ae^{3x^2+3x}$$


$$y=\frac{1-Ae^{6x^2+6x}}{1+Ae^{6x^2+6x}}$$


Something like this I think?