Integrating Partial Derivatives

[rudders93]

Homework Statement

Find the general function f(x,y) that satisifes the following first-order partial differential equations

$$\frac{df}{dx}=4x^3 - 4xy^2 + cos(x)$$
$$\frac{df}{dy}=-4yx^2 + 4y^3$$

The Attempt at a Solution

I integrated both to get:

$$x^4 - 2x^2y^2 + sin(x) + c(y)$$

and

$$-2y^2x^2 + y^4 + c(x)$$

I'm not to sure what it means by the most general function that satisifes both. Like I'm assuming that means it's just some function that I can sub into both partial derivatives and get the same result? If so, what is a systematic way of doing so?

Thanks!

 

[ideasrule]

Like I'm assuming that means it's just some function that I can sub into both partial derivatives and get the same result?

I don't know what you mean by "get the same result", but it's some function that satisfies both partial differential equations.

Let's focus on your first solution: $$x^4 - 2x^2y^2 + sin(x) + c(y)$$

We know that the second differential equation imposes some extra conditions on c(y), so try differentiating this solution with respect to y and comparing with the second differential equation to see what c(y) must be.

You might wonder why I chose the first solution instead of the second. That's because the two give the same answer. If they didn't, the system of equations would have no solution.