Power series

(x+1)y"-(2-x)y'+y=0 y(0)=2,y'(0)=-1 use power series methods to solve differential equation with given initial values

Classification of first order ordinary differential equations

In order to solve differential equations, it is helpful to classify them as belonging to one or more categories. In this entry we will consider three common classes of first order ordinary differential equations (ODEs): separable, exact and linear. We will show how each class is defined.

Growth rate

A bacteria culture starts with 760 bacteria and grows at a rate proportional to its size. After 2 hours there will be 1520 bacteria. Express the population after t hours as a function of t.

Mixture problem

A tank contains 1320 L of pure water. A solution that contains .01kg of sugar per liter enters a tank at the rate 3L/min. The solution is mixed and drains from the tank at the same rate. Find the amount of sugar after t minutes as a function of t.

Mixing problem

A tank contains 1320L of pure water.A solution that contains .o1kg of sugar per liter enters a tank at the rate 3L/min The solution is mixed and drains from the tank at the same rate. Solve for function of t So far I have the equation:

Exact equation

The following differential equation is exact. Find a function F(x,y) whose level curves are solutions to the differential equation: ydy-xdx=0 "F(x,y) such that the solutions are F(x,y)=c for an arbitrary constant c".

Explicit/implicit solutions to diff. equation

Find an explicit or implicit solutions to the differential equation: (x^2 + 4xy)dx + xdy = 0 "F(x,y) such that the solutions are F(x,y)=c for an arbitrary constant c".

carrying capacity problem

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 7000 . The number of fish doubled in the first year. dP/dt = rP(1- p/K) find an expression for the size of the population after t years by determining constant r.

Logistic model

A population obeys the logistic model. It satisfies the equation : dP/dt = 2/1300 P(13-P) for P>0 Find when P is increasing and decreasing.

carrying capacity

Suppose that a population develops according to the logistic equation: dP/dt = 0.15P - 0.003P^2 where t is measured in weeks. what is the carrying capacity?