The correct question is:
Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose differential, dF(x,y) gives the differential equation. That is, level curves F(x,y) = C are solutions to the differential equation:
dy/dx = (-3x^4 + 2y)/(-2x - 4y^2).
First rewrite as M(x,y)dx + N(x,y)dy = 0 where M(x,y) = -3x^4 + 2y, and N(x,y) = 2x + 4y^2).
If the equation is not exact, enter not exact, otherwise enter in F(x,y) as the solution of the differential equation
Answer:
F(x, y) = 2(x + y)
Explanation:
Given
M(x,y) = -3x^4 + 2y, and N(x,y) = 2x + 4y^2).
Suppose the differential equation is exact, then
∂N/∂x = ∂M/∂y
∂N/∂x = 2, and ∂M/∂y = 2
Therefore the equation is exact.
The solution F(x, y) can be obtained by integrating ∂M/∂y and ∂N/∂x
Integral of ∂M/∂y is 2y + f(x)
Integral of ∂N/∂x is 2x + f(y)
F(x, y) = 2(x + y).
For arbitrary constants f(x) = f(y) = 0