CHAPTER 2 2.4 Continuity

Seperable EquationsA separable equation is a first-order differential

equation in which the expression for dy / dx can be factored as a function od x times a function of y. In other words, it can be written in the form

dy/dx = g(x)f(y).

If f(y) 0, we could write dy/dx = g(x)/h(y), where h(y) = 1 / f(y). To solve this equation, we rewrite it in the differential form h(y)dy = g(x)dx. After we integrate both sides, we get: h(y)dy = g(x)dx.

CHAPTER 2 2.4 Continuity

Example: Solve the differential equation dy/dx = (x + sin x) / 3y2.

Example: Solve the differential equation dx/dt = 1 + t - x – t x.

Example: Solve the differential equation that satisfies the given initial condition. dx/dt = (1 + x) / (x y), x >0, y(1)=-4.

CHAPTER 2 2.4 Continuity

Example: Solve the differential equation that satisfies the given initial condition. dy/dt = (t y + 3t) / (t2 + 1), y(2)=2.

Example: Solve the differential equation that satisfies the given initial condition. x + 2y ((x2 + 1)) dy/dt = 0, y(0)=1.

Orthogonal Trajectories: An orthogonal trajectory of a family of curves is a curve that intersects each curve of the family orthogonally, that is, at right angles.

Example: Find the orthogonal trajectories of the family of curves.

a) x2 + y2 = k b) y = k e-x

CHAPTER 2 2.4 Continuity

Mixing Problems: A typical mixing problem involves a tank of fixed capacity filled with a thoroughly mixed solution of some substance (say, salt). A solution of given concentration enters the tank at a fixed rate and the mixture, thoroughly stirred, leaves at a fixed rate, which may differ from the entering rate. If y(t) denotes the amount of substance in the tank at time t, then y’(t) is the rate at which the substance is being added minus the rate at which it is being removed. The mathematical description of this situation often leads to a first-order separable differential equation.

CHAPTER 2 2.4 Continuity

Example: A tank contains 1000 L of pure water. Brine that contains 0.05kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04kg of salt per liter of water enters the tank at a rate of 10L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15L/min. How much salt is in the tank a) after t minutes b) after 1 hour?

CHAPTER 2 2.4 Continuity

Example: A certain small country has $10 billion in paper currency circulation, and each day $50 million comes into the country’s banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let x = x(t) denote the amount of new currency in circulation at time t, with x(0)=0.

a) Formulate a mathematical model in the form of an initial-value problem that represents the “flow” of the new currency into circulation.

b) Solve the initial-value problem found in part(a).

c) How long will it take for the new bills to account for 90% of the currency in circulation?