Exact Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Exact Equations

Method of Solving Exact Equations

  1. Identify M(x,y) from the exact differential form.

  2. Identify N(x,y) from the exact differential form.

  3. Find M(x,y)y.

  4. Find N(x,y)x.

  5. Compare M(x,y)y and N(x,y)x.

    • If M(x,y)y=N(x,y)x, then the differential equation is an exact equation, proceed to next step.

    • If M(x,y)yN(x,y)x, then we need another method to be discussed in a future class period.

  6. Since we know partial derivatives of F(x,y), we need to integrate the partial derivatives to find F(x,y):

    • Since M(x,y)=F(x,y)x, to find F(x,y) evaluate: M(x,y)dx, OR

    • Since N(x,y)=F(x,y)y, to find F(x,y) evaluate: N(x,y)dy

  7. Since we are evaluating indefinite integrals, the antiderivatives require a constant term:

    • When integrating with respect to x, y is a constant; evaluate F(x,y)=M(x,y)dx+g(y); OR

    • When integrating with respect to y, x is a constant; evaluate F(x,y)=N(x,y)dy+g(x)

  8. Find the constant of integration:

    1. Case 1

      • Evaluate: F(x,y)y=[M(x,y)dx+g(y)]y=M(x,y)+g(y) [which is equivalent to N(x,y)]

      • Solve M(x,y)+g(y)=N(x,y) for g(y)

      • g(y)=g(y)dy

    2. Case 2

      • Evaluate: F(x,y)x=[N(x,y)dy+g(x)]x=N(x,y)+g(x) [which is equivalent to M(x,y)]

      • Solve N(x,y)+g(x)=M(x,y) for g(x)

      • g(x)=g(x)dx

  9. Thus, F(x,y)=M(x,y)dx+g(y) OR F(x,y)=N(x,y)dy+g(x) usually, written as F(x,y)=C

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Last Modified: Wednesday, 2 September 2020 07:42 EDT