a) Determine whether the following statements are true or false and justify your response.

i. y=C₁e^3x+C₂e^-3x is the general solution of (d²y/dx²)-6(dy/dx)+9=0.

ii.y_p=Cx²e^2x is the particular solution of (d²y/dx²)+4y=3e^2x.

b) What is the characteristic equation and what role does it play in the solution to a linear, second-order difference and differential equations?

c) Consider a model of market equilibrium in which the current supply of firms is a function of the price that is expected to prevail when the product is sold. Assume that the market supply equation is

q²(t)=F+Gp^e

and is the expected price and and are constant parameters of the supply equation. Assume further that suppliers use information about the actual current price and its first and second derivatives with respect to time to form their prediction of the price that will prevail when their product reaches the market. In particular, assume that

p^e=p+b(dp/dt)+c(d²p/dt²)

and > > 0. If the current price is constant, so that, then suppliers expect the prevailing price to equal the current price. If the current price is rising, so that dp/dt > 0, then suppliers expect the prevailing price to be higher than the current price. How much higher depends on whether the current price is rising at an increasing rate, d²p/dt² > 0; or at a decreasing rate, d²p/dt² < 0.

The remainder of the market equilibrium model is a linear demand equation

q^D=A+Bp

and a linear price-adjustment equation that says that price rises when there is excess demand and falls when there is excess supply:

dp/dt = (q^D-q^S)

and > 0 is a constant which determines how rapidly price adjusts when the market is out of equilibrium and A and B are constant parameters of the demand equation.

i.Derive the linear second-order differential equation implied by this model.

ii. Given = 0.1, = 25, = 20, = 0.5 = 0.1, solve the differential equation.