6. Dynamic Analysis

Question 1

What is a differential equation?

Answer 1

An equation that relates the values of a function to its derivatives

Question 2

How can we classify the types of differential equations

Answer 2

-Order- the order of a differential equation is determined by the highest order of derivative contained in the equation
-Autonomous- autonomous differential equations don’t depend on time explicitly
-Linear or nonlinear- non linear differential equations involve non linear y terms

Question 3

What can we do to solve a differential equation if we can’t solve it explicitly

Answer 3

We can draw a phase diagram

Question 4

How can we split the problem of a linear first order autonomous differential equation into two?

Answer 4

We divide the problem into two:
-find a solution to homogeneous fork of equation e.g. ŷ+ay=0
-find a solution to particular form of equation e.g. the steady state solution satisfying the condition ŷ=0
The GeV set so solution is the sum of the solutions to these two problems

Question 5

What is the initial value problem?

Answer 5

When we know the initial value of the variable y ie the value of y at time zero

Question 6

What do we use an integrating factor for?

Answer 6

To make it possible to directly integrate the equation

Question 7

What does a phase diagram do?

Answer 7

It informs us of the behaviour of the solution. Identified values of y around which solution increases or decreases. Assesses the stability of steady state

Question 8

When is a non linear first order differential equation unstable?

Answer 8

If dŷ/dy>0

Question 9

When is a non linear first order differential equation stable?

Answer 9

If dŷ/dy<0 stable

Question 10

How do we get the complete solutions from the homogeneous and particular solutions?

Answer 10

Add the homogeneous and particular solutions together

Question 11

For systems of 2 equating how can stability be assessed?

Answer 11

By evaluating det(A) and trace(A) since det(A)=r1 x r2 and trace(A)=r1 + r2