Chapter 3.2 Existence and Uniqueness

Question 1

Exist. and Uniqueness of solution:

Answer 1

Assume there are intervals for x1,x2,x3… for their initial value conditions where each variable’s functions are continuous, and each partial derivative is also continuous. So, for 2d that is 6 functions to ensure are cont.

Question 2

What if one partial derivative is not continuous at a certain t? Then,

Answer 2

Then, as long as your initial value is not at that point, then there exists a unique soln satisfying IC.

Question 3

Autonomous means for a system of equations:

Answer 3

None of the equations involve t

Question 4

How to draw a slope field?

Answer 4

Come up with equation for vector in terms of slope. i.e. the x’ and y’ in terms of the x,y. Then, plug in all the points.

Question 5

What is an x-nullcline?

Answer 5

(In a slope field typically) is the collection of curves satisfying x’=0 AKA the slope is UP or DOWN. Same for y-nullclines.

Question 6

What is the property of passing through a nullcline?

Answer 6

When you pass through a nullcline, x’,y’ goes from positive to negative.