The net profit p and quantity x satisfy the differential equation \(\frac { dp }{ dx } =\frac { 2{ p }^{ 3 }-{ x }^{ 3 } }{ 3x{ p }^{ 2 } } \). Find the relationship between the net profit and demand given that p = 20, when x = 10.

The rate of increase in the cost Cof ordering holding as the size q of the order increases is given by the differential equation \(\frac { dc }{ dq } =\frac { { c }^{ 2 }+2cq }{ { q }^{ 2 } } \). Find the relationship between c and q if c = 1 when q = 1.

The total cost of production y and the level of output x are related to the marginal cost of production by the equation (6x²+2y²)dx-(x²+4xy)dy = 0. What is the relation between total cost and output if y = 2 when x = 1?

Equipment maintenance and operating costs (are related to the overhaul interval x by the equation \({ x }^{ 2 }\frac { dc }{ dx } -10xc=-10\) with c = c₀ and x = x₀. Find c as a function of x.

Suppose that the quantity needed Q_d = 42 -4p-4\(\frac { dp }{ dt } +\frac { { d }^{ 2 }p }{ { dt }^{ 2 } } \) and quantity supplied Q_s = -6 + 8p where p is the price. Find the s equilibrium price for market clearance.