Exact Equations

Posted on May 1, 2015 by | 1 Comment

Exact Equation is a method of solving first order differential equations. First order differential equations come in different forms and there are many different methods for solving. One example of a first order differential equation will generally contain constants, an initial parameter and most importantly containing only 1 variable. Exact equation question consist of all that IVP contains however there are 2 unknown variables. A hint that the question tells you to approach the question using exact equation is when the equation is stated as df=(df/dx) dx +(df/dy) dy

For example:   df=(8x+9x^2y^3)+(9x^3y^2+cos(y)y')

8x+9x^2y^3 as My and 9x^3y^2+cos(y) as Nx

The steps to take are:

1.Test for Exactness of the equation

2.Take derivative of My with respect to y

8x+9x^2y^3 dy = 27x^2y^2

3.Take derivative of Nx in respect to x

9x^3y^2+cos(y)dx = 27x^2y^2

If the derivatives come out to be equal then the problem can be solved with the method of exact equation.

4.Taking integration of My with respect to x

\int 8x+9x^2y^3 dx = 4x^2+3x^3y^3+h(y)

5. Derive the answer from the previous part with respect to y where x will be consider a constant

4x^2+3x^3y^3+h(y) dy = 9x^3y^2+h'(y)

6.Set it equal to Nx equal to the answer from previous part and use algebra to take out parts of the equation that are equal to each other in order to find h'(y).

9x^3y^2+h'(y) = 9x^3y^2+cos(y)

As we can see 9x^3y^2 will get crossed out from both side of the equation when it is subtracted from one side to the other leaving h'(y) equal to cos(y).

7.Solve for h(y) so it can be placed back into the equation of 2i. In order to solve we must take the integration of h'(y) with respect to y.

\int h'(y) = \int cos(y) h(y) = sin(y)+c

8.Place h(y) back into the equation at 2i and we will get :

4x^2+3x^3y^3+sin(y) = c

I have included some useful materials on how to solve exact equation. These video has helped me with my homework as well as teaching me the material clearly in order for me to answer the exact equation question correctly on the test; therefore I highly recommend checking these videos out.

First video: Exact Differential Equations from patrickJMT of Youtube.

Second video: Exact differential equations:how to solve from Dr Chris Tisdell

 

If you have any more questions about exact equations after watching the videos, please go meet with your professor to get additional help. Thank you!

Resources:

http://www.cliffsnotes.com/math/differential-equations/first-order-equations/exact-equations

Example taken from Professor Reitz Spring 2015 Class.

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