Please help with 1st-order equations

Question

From below choices, select all 1st-order equations that can be classified as an Exact ODE.

A.)

B.)

C.)

D.)

Answer 1

A.) is exact equation because ∂/∂y(1/x + y) = ∂/∂x(1/y + x) = 1;

B.) is exact because ∂/dy(f(x)) = ∂/∂x(- 1) = 0

C.) is not exact because ∂/dy(2ysinx) = 2sinx ≠ ∂/dx(y²cosx) = - y²sinx

D.) is not exact because ∂/dy(x²y) = x² ≠ ∂/dx(y²x) = y²

Answer 2

Given P(x,y)dx+Q(x,y)dy=0, for exactness you have to check the condition: ∂P/∂y=∂Q/∂x.



A) ∂P/∂y=1=∂Q/∂x => exact

B) ∂P/∂y=0=∂Q/∂x => exact

C) ∂P/∂y=2*sin(x), ∂Q/∂x=-y²sin(x) => not exact

D) ∂P/∂y=x², ∂Q/∂x=y² => not exact