Exact Equation

 

The differential equation     Mdx+Ndy=0    ,

where both   M   and   N   are functions of    x   and   y     is said to be exact when there is a function     u     of     x, y    such that 

Mdx+Ndy=du ,

i.e. ,  when   Mdx+Ndy   becomes a partial differential .

Now , we know from Differential Calculus that ;

Mdx+Ndy   should be a perfect differential if ;

DM/Dy=DN/Dx  ,   Hence the condition that

Mdx+Ndy=0  should be an exact differential equation is

DM/Dy=DN/Dx

The method of solving and exact equation of the type

Mdx+Ndy=0   .

First integrate the term in   Mdx   as if   y   were constant then integrate the terms in   Ndy   considering    x   as constant and rejecting the terms already obtained equate the sum of these integrals to a constant .