Solve the equation separable, linear, bernoulli, or homogenous1.dydx=x4+4xy22x3+x2y+y32.(e−ycos⁡(x))y′=x4+6x2y33.y′=y+yx3x+x2cos⁡(x2y2)

Separable equation:
A first order differential equation y’=f(x,y) is called a separable equation if the function f(x,y) can be factored into the product of two functions of x and y
${\displaystyle f(x,y)=f_1\left(x\right)f_2\left(y\right)}$
where ${\displaystyle f_1\left(x\right)}$ and ${\displaystyle f_2\left(y\right)}$ are continuous functions.
Linear differential equation:
A first order differential equation is linear when it can be written as:
${\displaystyle \frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)}$
Where P(x) and Q(x) are functions of x.
Bernoulli differential equation:
A first order Bernoulli differential equation can be written as: ${\displaystyle \frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)}$
Where P(x) and Q(x) are functions of x.
Bernoulli differential equation:
A first order Bernoulli differential equation can be written as: ${\displaystyle \frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)y^n}$
Where P(x) and Q(x) are functions of x.
Homogeneous differential equation:
A first order homogeneous differential equation can be written as: ${\displaystyle \frac{dy}{dx}=F\left(\frac{y}{x}\right)}$
All three equations are written in the table :
\[\begin{array}{|c|c|}\hline Equation & Separable & Linear & Bernoilli & Homogeneous\ \hline \frac{dy}{dx}=\frac{x^4+4xy^2}{2x^3+x^2y+y^3} & no & no & no & no\\hline (e^{-y}\cos(x))y=x^4+6x^2y^3