Determine if the given differential equation is exact. If it is exact solve it.

Question

Answer 1

Because ∂/∂y(x - y³ + y²sinx) = - 3y² + 2ysinx = ∂/∂x(- 3xy² - 2ycosx).

Then ∂F(x,y)∂/x = (x - y³ + y²sinx); F(x,y) = ∫(x - y³ + y²sinx)dx = x²/2 - y³x - y²cosx + f(y);

∂F(x,y)/∂y = -3y²x - 2ycosx + f'(y) = - 3xy² - 2ycosx; so f'(y) = 0 and f(y) = C

So solution of equation: x²/2 - y³x - y²cosx = C