Determine whether a differential equation is exact or not can save you time if you have a lot of differential equations to deal with because you don't need to waste time trying to find the function f(x, y) if an equation isn’t exact. Here is one example on how to do it, and other questions will just be similar in steps:

Example: Is the differential equation 6x+y^2+6xy dy/dx = 0 exact?

Answer: We know that to determine whether a differential equation is exact or not we first need to put it into the form M (x,y)+N(x,y) dy/dx=0.

So let's do it:

In this example we have M(x,y)=6x+y^2 and N(x,y)=6xy

We can then attain `del` M(x,y)/`del` y=6y, and `del` N(x,y)/`del` x=6y

Not difficut to see that `del` M(x,y)/` ` y=`del` N(x,y)/` ` x, and therefore the differential equation 6x+y^2+6xy dy/dx = 0 exact

 

(Note: Remember, `del` M(x,y)/` ` y=`` N(x,y)/` ` x is the key to determine it is exact or not)