C4 and C5

Question 1

Definition: Direction Fields

Answer 1

Consider the ODE y’ = f(t,y). We can draw a direction field as follows:

Draw a t-y coordinate system.
Evaluate f(t,y) over a rectangular grid of points.
Draw a line at each point (t,y) of the grid with slope f(t,y)

Question 2

Why are direction fields and phase plots useful?

Answer 2

But even without solving it, can we answer this question:

If y(0)=0, what will be the long term behaviour of the solution? Will it increase/decrease? Will it go to infinity?But even without solving it, can we answer this question with direction fields and phase plots

Question 3

Definition: Phase Plots

Answer 3

Consider an autonomous ODE y’ = f(y) . The graph of f(y) in the y - y’ plane is called the phase plot of this ODE.

Question 4

What are the steps to solving an initial value problem? What are some things to look out for?

Answer 4

First step: Find the rate at which salt (or anything else) enters the tank (or anything else)
rate in = concentration x flow rate

Second step: Find the rate at which salt leaves the tank
rate out = concentration x flow rate
concentration = Q(t)/volume, where Q(t) is the amount of the substance at time t

Third step: Write down and solve the differential equation
where Q’(t) = rate in - rate out
where rate out depends on Q(t) . Now solve for Q(t) using integrating factor or through separable ODE

Fourth step: Answer the questions from the beginning.

BE CAREFUL OF:
- times in which the situation doesn’t make sense (tank overflows at one point, cannot hold some amount etc) in which we have to find when it stops making sense, cut off from there and use that as the new initial condition for the cut off after

• buffers

remember, every situation is different. so best to practice to become familiar