Question

Solve the equation
`(12 x^(5)y+12 x^(6)y-3 y^(5) -18x y^(5)) dx + (2 x^(6)-15x y^(6) )dy=0` leaving your answer in implicit form.

Answer 1

`\underbrace{(12x^5y+12x^6y-3y^5-18xy^5)}_M\,\mathrm dx+\underbrace{(2x^6-15xy^4)}_N\,\mathrm dy=0`


`M_y=12x^5+12x^6-15y^4-90xy^4`

`N_x=12x^5-15y^4`


`-\frac{N_x-M_y}N=6\implies\mu(x)=\exp\left(\displaystyle\int6\,\mathrm dx\right)=e^(6x)`


`\underbrace{(12x^5y+12x^6y-3y^5-18xy^5)e^(6x)}_(\mu M)\,\mathrm dx+\underbrace{(2x^6-15xy^4)e^(6x)}_(\mu N)\,\mathrm dy=0`

You can verify that the partial derivatives are equal.


`F_x=\mu M`

`F=(2x^6y-3xy^5)e^(6x)+f(y)`


`F_y=\mu N`

`(2x^6-15xy^4)e^(6x)+f'(y)=(2x^6-15xy^4)e^(6x)`

`f'(y)=0`

`\implies f(y)=C`


`\implies F(x,y)=(2x^6y-3xy^5)e^(6x)=C`