Answer
a) $b=c$
b) $c=-2b$
Work Step by Step
a) For a vector field $F$ to be Conservative, we must have $\dfrac{\partial F_1}{\partial y}=\dfrac{\partial F_2}{\partial x}$
We are given that the force field as: $F(x,y)=(ax+by, cx+dy)$
Here, we have
$\dfrac{\partial F_1}{\partial y}=b$ and $\dfrac{\partial F_2}{\partial x}=c$
This means that $b=c$ for a vector fields $F$ to be conservative.
b) For a vector field $F$ to be Conservative, we must have $\dfrac{\partial F_1}{\partial y}=\dfrac{\partial F_2}{\partial x}$
We are given that the force field as: $F(x,y)=(ax^2-by^2, cxy)$
Here, we have
$\dfrac{\partial F_1}{\partial y}=-2by$ and $\dfrac{\partial F_2}{\partial x}=cy$
This means that $c=-2b$ for a vector fields $F$ to be conservative.
