Test 1 Prep Flashcards
T or F: all models are DE’s
false
f’ (c)<0 means what?
f is decreasing at x = c
f’ (c)>0 means what?
f is increasing at x = c
What is a DE?
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)
what is an ODE (Ordinary DE)?
a DE involing ONLY ordinary derivatives w/ respect to a single indepednedent variable
what is PDE (Partial DE)?
a DE involing partial derivatives w/ respect to a single indepednedent variable
how do you determine the oder of DE?
Look at the highest derivative in the DE. Ex. If the highest is first derivative, the order of the DE, First order.
how do you determine independent and dependent variables.
dy/dx. derative of y W/ RESPECT TO X
derivative of (dep var.) W/ RESPECT TO (indep variable)
Linear vs Non Linear DE
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)
how does implicit differtiation work?
Basically chain rule everything. Leave primes except for variable were deriving with respect to
how to check something is a solution?
Plug it in and check if LHS = RHS.
Explicit vs implicit functions
A function is explicit when we can write is “y = some function of x”. It’s implicit when we don’t see that.
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
explicit solution
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.
implicit solution
difference b/w interval vs domain
intervals have to be a an connect set, domains can have breaks or unions.
What is an initial value problem (IVP)?
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
how to use existence thrm?
- write DE in proper form: y’ = f(x,y)
- Identify all discontinuities in f
- 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.
- If you can do step 3 then according to the existence theorem, a solution to the IVP exists
- if you can’t do step 3 then no conculsion can be drawn. IVP may or may not have a sol-n.
how to use existence & uniqueness thrm
- write in proper form. y’ = f(x,y) (y’ with a coeffeicent of one)
- compute partial derivative of the dependent variable. ∂f/∂y
- identify all discontinuities in f and ∂f/∂y
- 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.
- if you can do step 4, then according to the thrm a unique solution to the IVP exists.
- If you cannot do step 4, then there may or may not be a unique solution to IVP. We know nothing!
why do we care abt existence thrm?
if the DE model, doesn’t have a solution, then it’s a bad model and we can’t predict the future.
why do we care abt the existence & uniqueness thrm?
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.
how to find singular sol-ns?
- upon separating, look for any possible values of y (since y is the sol-n) that cause us to divide by zero.
- 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.
