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## Example of Exact Equations
Solve the Equation

![](https://i.imgur.com/D9rKbdA.png)




Let:
> []

![](https://i.imgur.com/HskvmIb.png)


Consider the First Order Partials
![](https://i.imgur.com/nFnU9vy.png)
Thus the equation is exact, and so we know that there must exist some function $f(x,y) = c$ such that:

![](https://i.imgur.com/KOvXBro.png)

So first partially integrate $M(x,y)$ in order to find the class of function corresponding to $f(x,y)$:
![](https://i.imgur.com/fK59mgE.png)

Now Partially Differentiate $f(x,y)$ to get $N(x,y)$:
![](https://i.imgur.com/RjLIcD5.png)

Now Equate the Terms:


![](https://i.imgur.com/wUHC2ee.png)

State $f(x,y)$:

![](https://i.imgur.com/s7CZ1L2.png)

This is merely an implicit solution, but ideally we would want an explicit solution:



![](https://i.imgur.com/ljpX3kJ.png)



