Question

Solve` (2y^2-4x+5)dx=(y-2y^2-4xy)dy` 

Answer 1

` (2y^2-4x+5)dx=(y-2y^2-4xy)dy`  

Compare with Mdx + Ndy = 0 

∴ M = (2𝒚^𝟐−𝟒𝒙+𝟓)                      ∴ N = - (𝒚−𝟐𝒚^𝟐−𝟒𝒙𝒚)

`(delM)/(dely)=4y`                           `(delN)/(del x)=4y`  

` ∴ (delM)/(dely)=(delN)/(del x)` 

The given diff. eqn is exact .
The solution of exact diff. eqn is given by , 

`int Mdx + int[N-del/(del y) Mdx]dy=c`

`int= int((2y^2-4x+5)) dx=2xy^2-2x^2+5x`

`del/(del y) intMdx=4xy` 

`int[N-del/(dely) Mdx]dy=int[4xy-y+2y^2-4xy]dy=2/3y^3-y^2/2` 

∴ `2xy^2-2x^2+5x+2/3y^3-y^2/2=c`