Exact Equations | Equations of Order One

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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.