show that the differential equation (x^2 cos y + y^2) + dy + 2x sin ydx = 0 is exact and find its solution

Question

show that the differential equation (x^2 cos y + y^2) + dy + 2x sin ydx = 0 is exact and find its solution

Answer 1

∂/∂x(x²cosy + y²) = ∂/∂y(2xsiny) = 2xcosy. So, this is exact equation

∂F/∂x = 2xsiny; F(x,y) = ∫2xsinydx = x²siny + f(y);

∂F/∂y = x²cosy + f'(y) = x²cosy + f'(y) = x²cosy + y²; f'(y) = y²; f(y) = y³/3 + C.

So, solution is F(x, y) = x²siny + y³/3 + C.