Exact Equations

An exact equation is a first order differential equation. The steps to solving an exact equation is to check if M_Y  is equal to Nx. to check if they are equal you have to derive by M by y and derive N by x. the next step is to Integrate M with respect to X and obtain  and have to make sure to add the h(y) or else you will get an error later on. next take the derivative of  what the step done before but with respect to y.next you have to it equal to N and make sure that h(y) was also derived in the previous step. next solve for h'(y). after solving h'(y) we now integrate h'(y) to find h(y). After this we plug in h(y) to to the step where we integrated M with respect to x and make sure the equation is equal to C.

Next I will show an example for Exact equations:

finding the general solution for .

the first step is to derive  with respect y and  with respect to x.

For  we get 2x and for  we get 2x as well so they are both equal to each other.

the next step is to integrate M with respect to y.

.

after integrating you should have gotten .

the next step is deriving what you just integrated with respect to y.

when deriving you should get .

next make  equal to N which is .

the next step would be to find h'(y).

h'(y) would be .

next we integrate h'(y).

after integrating we get

.

after finding h(y) we plug it into the equation we got when we integrated M with respect to y.

the equation should now be .

here are  videos on how to do exact equations if you need a better understanding:

https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/exact-equations/v/exact-equations-example-1

Resources:

http://tutorial.math.lamar.edu/Classes/DE/Exact.aspx

class notes