# Which Differential Equation Method Do I use?

#### lanczlot

i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.

#### mr fantastic

MHF Hall of Fame
i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.
Are you sure it's not (3y^2 - xy) dx -(x^2 + 6xy^2) dy=0?

#### lanczlot

the problem was hand written, so it may have been a mistake. what if it was squared instead of cube? how do you solve it?

MHF Hall of Fame

#### Jester

MHF Helper
i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.
If you write the ODE as

$$\displaystyle \frac{dy}{dx} = \frac{3y^3-xy}{x^2 + 6xy^2}$$

multiply both side by 2y

$$\displaystyle 2y\frac{dy}{dx} = \frac{6y^4-2xy^2}{x^2 + 6xy^2}$$

and let $$\displaystyle u = y^2$$ then

$$\displaystyle \frac{du}{dx} = \frac{6u^2-2xu}{x^2 + 6xu}$$ (this is homogeneous).

lanczlot

#### lanczlot

wait... how is the differential equation with u homogeneous?

#### Jester

MHF Helper
Divide everything on the rhs by $$\displaystyle x^2$$ so

$$\displaystyle \frac{du}{dx} = \frac{6\dfrac{u^2}{x^2}-2\dfrac{u}{x}}{1 + 6\dfrac{u}{x}}$$

which of the form is $$\displaystyle \frac{du}{dx} = f\left(\frac{u}{x}\right)$$.

#### Krizalid

MHF Hall of Honor
the ODE also admits an integrating factor of the form $$\displaystyle u(x,y)=x^my^n.$$