Use the \"mixed partials\" check to see if the following differential equation i

Question

Use the "mixed partials" check to see if the following differential equation is exact.

If it is exact find a function F(xy) whose differential, dF(xy) gives the differential equation. That is, level curves F(xy)=C are solutions to the differential equation:

dy/dx=(?x^(2)+2y) / (?2x?y)
First rewrite as

M(xy)dx+N(xy)dy=0

where M(xy)= ................... ,
and N(xy)=..................................... .

If the equation is not exact, enter not exact, otherwise enter in F(xy) as the solution of the differential equation here ............................ =C.

Explanation / Answer

dy/dx=(x^(2)+2y) / (2xy)

(-2x-y)dy = (-x2 + 2y)dx

(-x2+2y)dx + (2x+y)dy = 0

M(x,y) = (-x2+2y)

N(x,y) = 2x+y

dM/dy = 2

dN/dx = 2 = dM/dy -> The equation is exact.

We have:

int 2x+y dy = 2xy + 1/2 y2 + f(x)

d/dx (2xy + 1/2 y2 + f(x)) = 2y + f'(x) = (-x2+2y)

f'(x) = -x2 -> f(x) = -1/3 x3 + C

So the final solution is:

2xy + 1/2 y2 -1/3 x3 = C

F(x,y) = 2xy + 1/2 y2 -1/3 x3

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