Exact Equations | Equations of Order One

SPONSORED LINKS

The differential equation
 

$ M(x, y) \, dx + N(x, y) \, dy = 0 $

 

is an exact equation if
 

$ \dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x} $

 

Steps in Solving an Exact Equation

  1. Let $ \, \dfrac{\partial F}{\partial x} = M \, $.
  2. Write the equation in Step 1 into the form
     
    $ \displaystyle \int \partial F = \int M \, \partial x $

     

    and integrate it partially in terms of x holding y as constant.

  3. Differentiate partially in terms of y the result in Step 2 holding x as constant.
  4. Equate the result in Step 3 to N and collect similar terms.
  5. Integrate the result in Step 4 with respect to y, holding x as constant.
  6. Substitute the result in Step 5 to the result in Step 2 and equate the result to a constant c.

 

SPONSORED LINKS