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ivanzaharov [21]
3 years ago
6

Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function F(x,y) who

se level curves are solutions to the differential equation. (−3xy^2+y)dx + (−3x^2y+x)dy =0
Mathematics
1 answer:
Arturiano [62]3 years ago
3 0

We have

(-3xy^2+y)_y=--6xy+1

and

(-3x^2y+x)_x=-6xy+1

so the equation is indeed exact. So we want to find a function F(x,y)=C such that

F_x=-3xy^2+y

F_y=-3x^2y+x

Integrating both sides of the first equation wrt x gives

F(x,y)=-\dfrac32x^2y^2+xy+f(y)

Differentiating both sides wrt y gives

F_y=-3x^2y+x=-3x^2y+x+f_y\implies f_y=0\implies f(y)=C

So we have

F(x,y)=-\dfrac32x^2y^2+xy+C=C

or

F(x,y)=\boxed{-\dfrac32x^2y^2+xy=C}

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