Test 1 Prep

Question 1

T or F: all models are DE’s

Answer 1

false

Question 2

f’ (c)<0 means what?

Answer 2

f is decreasing at x = c

Question 3

f’ (c)>0 means what?

Answer 3

f is increasing at x = c

Question 4

What is a DE?

Answer 4

A DE is an equation involving a dependent variable (unknown function) and one or more of it’s derivaties with respect to the independent variable(s)

Question 5

what is an ODE (Ordinary DE)?

Answer 5

a DE involing ONLY ordinary derivatives w/ respect to a single indepednedent variable

Question 6

what is PDE (Partial DE)?

Answer 6

a DE involing partial derivatives w/ respect to a single indepednedent variable

Question 7

how do you determine the oder of DE?

Answer 7

Look at the highest derivative in the DE. Ex. If the highest is first derivative, the order of the DE, First order.

Question 8

how do you determine independent and dependent variables.

Answer 8

dy/dx. derative of y W/ RESPECT TO X
derivative of (dep var.) W/ RESPECT TO (indep variable)

Question 9

Linear vs Non Linear DE

Answer 9

A DE is linear if the depedent variable and tis’ derivatives appear in additive combinations of their first power. (No Multiplying variables. Not to the power of a the independent variable)

Question 10

how does implicit differtiation work?

Answer 10

Basically chain rule everything. Leave primes except for variable were deriving with respect to

Question 11

how to check something is a solution?

Answer 11

Plug it in and check if LHS = RHS.

Question 12

Explicit vs implicit functions

Answer 12

A function is explicit when we can write is “y = some function of x”. It’s implicit when we don’t see that.

Question 13

For a DE with dependent varaible y and independent varaible x, an ______ to the DE on an interval I is a function y=f(x) that satifies the equation for all x ∈ I

Answer 13

explicit solution

Question 14

A relation G(x,y) is said to be an _______ to a DE on an interval I provided that there exists at least one function f that satisfies the relation as well as the DE on I.

Answer 14

implicit solution

Question 15

difference b/w interval vs domain

Answer 15

intervals have to be a an connect set, domains can have breaks or unions.

Question 16

What is an initial value problem (IVP)?

Answer 16

an IVP for an nth order DE is one in which the n inital conditions on the solution y and it’s derivatives must be satisfied:

y(x_0) = y_0
y’(x_0) = y_1
y^(n-1) (x_0) = y_n-1

Question 17

how to use existence thrm?

Answer 17

1. write DE in proper form: y’ = f(x,y)
2. Identify all discontinuities in f
3. make a rectangle containging (x_0, y_0) on the inside that also has none of the points where f is discontinuous. You really want to make sure that there is no discontinuity at the IC.
4. If you can do step 3 then according to the existence theorem, a solution to the IVP exists
5. if you can’t do step 3 then no conculsion can be drawn. IVP may or may not have a sol-n.

Question 18

how to use existence & uniqueness thrm

Answer 18

1. write in proper form. y’ = f(x,y) (y’ with a coeffeicent of one)
2. compute partial derivative of the dependent variable. ∂f/∂y
3. identify all discontinuities in f and ∂f/∂y
4. try to construct a rectangle containing (x_0, y_0) on the inside that also has none of the points where f or ∂f/∂y is discontinuous.
5. if you can do step 4, then according to the thrm a unique solution to the IVP exists.
6. If you cannot do step 4, then there may or may not be a unique solution to IVP. We know nothing!

Question 19

why do we care abt existence thrm?

Answer 19

if the DE model, doesn’t have a solution, then it’s a bad model and we can’t predict the future.

Question 20

why do we care abt the existence & uniqueness thrm?

Answer 20

if a model has more than one sol-n, which is when the thrm fails. We don’t know which sol-n correctly predicts the future.

Question 21

how to find singular sol-ns?

Answer 21

1. upon separating, look for any possible values of y (since y is the sol-n) that cause us to divide by zero.
2. for any such y (ȳ), ask yourself if it is possible to find a constant in the one parameter family that allows you to find a sol-n that sovles the IC for all x_0 in the sol-ns interval.
   a. If you can get ȳ from the your general sol-n then ȳ isn’t a signular solution
   b. If you can’t, y = ȳ is a singular solution.