Consider the IVPs:
(A) y'+2y = 1, 0<t<1 , y(0)=2.
(B) y' = y(1-y), 0<t<1 , y(0)=1/2.
1. For each one, do the following:
a. Find the exact solution y(t) and evaluate it at t=1.
b. Apply Euler's method with Δt=1/4 to find Y4 ≈ y(1).
Make a table of tn, Yn for n=0,1,2,3,4.
c. Find the error at t=1.
2. Euler's method is obtained by approximating y'(tn) by a forward finite difference.
Use the backward difference approximation to y'(tn+1) to derive the
Backward Euler Method: Yn+1 = Yn + Δt f(tn+1, Yn+1) , n=0,1,2,...
Note that now the unknown Yn+1 appears inside f(.,.), so this equation needs to be
solved for Yn+1 at each time-step!!! whence it is also called Implicit Euler Method.
For the simple ODEs (A), (B) above, the updating equation can be solved by hand,
but in general a root-finder (like Newton-Raphson) is needed.
This scheme is also 1st order, but it has better stability properties than Explicit Euler.
For each IVP problem (A), (B), do the following:
a. Apply the Backward Euler Method with Δt=1/4 to find Y4 ≈ y(1).
b. Find the error at t=1 and compare with Explicit Euler.
3. Use the centered difference approximation to y'(tn) to derive the so called
Midpoint Method: Yn+1 = Yn-1 + 2 Δt f(tn, Yn) , n=1,2,...
Note that this requires both Yn-1 and Yn to produce Yn+1. It is a 2-step method,
hence not self-starting (need Y0 and Y1 before it can be applied),
so some single-step method (like Euler) must be used to start it off.
However, it has the advantage of being a 2nd order method, and explicit.
For each IVP problem (A), (B), do the following:
a. Apply the Midpoint Method with Δt=1/4 to find Y4 ≈ y(1).
b. Find the error at t=1 and compare with Explicit Euler and with Implicit Euler.
Which method seems to be doing better in this case ?In: Advanced Math
the table below lists information for seven Cepheid variables where ‘Period’ is the pulsation period of the star, and ?/?⊙ is the luminosity in solar units.
|
Period (days) |
L/L⨀ |
|
4.5 |
835 |
|
7.2 |
1418 |
|
15.0 |
3761 |
|
29.3 |
8176 |
|
51.6 |
16623 |
|
82.8 |
28220 |
|
172.5 |
75140 |
a) Plot the PL relation for these stars using the logarithms of the values in the table (that is, ???(?) vs ???(?/?⊙)) and extract the slope and y-intercept of the relation the data define
In: Advanced Math
Maximize p = 13x + 8y subject to
| x | + | y | ≤ | 25 | |||
| x | ≥ | 10 | |||||
| −x | + | 2y | ≥ | 0 | |||
| x ≥ 0, y ≥ 0. | |||||||
P = ? (X,Y)= ( ?,? )
In: Advanced Math
Let T denotes the counterclockwise rotation through 60∘, followed by reflection in the line y=x.
(i) Show that T is a linear transformation.
(ii) Write it as a composition of two linear transformations.
(iii) Find the standard matrix of T.
In: Advanced Math
If we recreated the scene from Fast & Furious 7 and dropped
a Challenger SRT® Hellcat Redeye Widebody from a C-130 aircraft at
5,280 ft, how much horsepower would it take to drive past it before
it hits the ground if you’re 1 mile away?
Pro Tips
Air density @ sea level, 59 degrees, no wind = p =
.002377 slugs/ft^3
Coefficient of drag (flat plate, NASA) = C(d) =
1.28
Weight = W = 4451 lbs
Gravitation constant = g = 32.2 ft/sec^2
Area = A = 197.5" long x 78.2" wide x (1 ft^2/ 144
in^2)
Vehicle falls flat, wheels 1st, straight down, at
constant acceleration with no aerodynamic drag until terminal
velocity
Horsepower needed to accelerate is AVERAGE - not
peak
100% driveline efficiency
In: Advanced Math
Patio Iron makes wrought iron outdoor dining tables, chairs, and stools. Each table uses 8 feet of a standard width wrought iron, 2 hours of labor for cutting and assembly, and 2 hours of labor for detail and finishing work. Each chair uses 6 feet of the wrought iron, 2 hours of cutting and assembly labor, and 1.5 hours of detail and finishing labor. Each stool uses 1 foot of the wrought iron, 1.5 hours for cutting and assembly, and 0.5 hour for detail and finishing work, and the daily demand for stools is at most 10. Each day Patio Iron has available at most 154 feet of wrought iron, 55 hours for cutting and assembly, and 50 hours for detail and finishing. If the profits are $60 for each dining table, $48 for each chair, and $36 for each stool, how many of each item should be made each day to maximize profit?
In: Advanced Math
Given two planes 2x - y + z = 7 and x + 3y - 4z = 1.
(a) Give an orthogonal vector to each plane.
(b) Do the planes intersect? Why or why not?
(c) If they intersect, find the parametric equation of the intersection line, if not, find the distance of both planes.
In: Advanced Math
Let G be a simple undirected graph with n vertices where n is an even number. Prove that G contains a triangle if it has at least (n^2 / 4) + 1 edges using mathematical induction.
In: Advanced Math
Use induction to prove that for any positive integer n, 8^n - 3^n is a multiple of 5.
In: Advanced Math
1. Find the quotient and remainder when 74 is divided by 13.
2. Use the Euclidean Algorithm to find the GCD of 201 and 111.
3. Express your answer to #2 as a combination of 201 and 111.
4. In Z7 compute the following: a. 4+6, b. 4. 6, c. 35.
In: Advanced Math
For the given functions f and g, complete parts (a)-(h). For parts (a)-(d), also find the domain.
f left parenthesis x right parenthesis equals StartFraction 7 x plus 9 Over 9 x minus 7 EndFractionf(x)=7x+99x−7;
g left parenthesis x right parenthesis equals StartFraction 9 x Over 9 x minus 7 EndFractiong(x)=9x9x−7(a) Find
(fplus+g)(x).
(fplus+g)(x)equals=nothing
(Simplify your answer.)What is the domain of
fplus+g?
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.The domain is
StartSet x vertical line nothing EndSetx .
(Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B.The domain is
StartSet x vertical line x is any real number EndSet{x x is any real number}.
(b) Find
left parenthesis f minus g right parenthesis left parenthesis x right parenthesis(f−g)(x).
left parenthesis f minus g right parenthesis left parenthesis x right parenthesis(f−g)(x)equals=nothing
(Simplify your answer.)What is the domain of
f minus gf−g?
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.The domain is
StartSet x vertical line nothing EndSetx .
(Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B.The domain is
StartSet x vertical line x is any real number EndSet{x x is any real number}.
(c) Find
(ftimes•g)(x).
(ftimes•g)(x)equals=nothing
(Simplify your answer.)What is the domain of
ftimes•g?
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.The domain is
StartSet x vertical line nothing EndSetx .
(Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B.The domain is
StartSet x vertical line x is any real number EndSet{x x is any real number}.
(d) Find
left parenthesis StartFraction f Over g EndFraction right parenthesis left parenthesis x right parenthesisfg(x).
left parenthesis StartFraction f Over g EndFraction right parenthesis left parenthesis x right parenthesisfg(x)equals=nothing
(Simplify your answer.)What is the domain of
StartFraction f Over g EndFractionfg?
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.The domain is
StartSet x vertical line nothing EndSetx .
(Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B.The domain is
StartSet x vertical line x is any real number EndSet{x x is any real number}.
(e) Find
(fplus+g)(44).
(fplus+g)(44)equals=nothing
(Type an integer or a simplified fraction.)(f) Find
(fminus−g)(33).
(fminus−g)(33)equals=nothing
(Type an integer or a simplified fraction.)(g) Find
(ftimes•g)(22).
(ftimes•g)(22)equals=nothing
(Type an integer or a simplified fraction.)(h) Find
left parenthesis StartFraction f Over g EndFraction right parenthesis left parenthesis 2 right parenthesisfg(2).
left parenthesis StartFraction f Over g EndFraction right parenthesis left parenthesis 2 right parenthesisfg(2)equals=nothing
(Type an integer or a simplified fraction.)
In: Advanced Math
A firm faces the demand schedule as ? = 660 − 3?
and the total cost schedule as
?? = 6?^3 − 72?^2 + 240? + 25. Please answer the followings:
a. Does the above cost function satisfy the parametric restrictions
we derived in the class?
b. What is the maximum profit the firm can make? Confirm your
results with second order
conditions as well.
In: Advanced Math
Use the Laplace transform to solve the given system of differential equations. dx/dt + 3x + dy/dt = 1 dx/dt − x + dy/dt − y = e^t x(0) = 0, y(0) = 0
In: Advanced Math
let a be a non zero constant and consider: y''+(1/t)y'=a
a. show that 1 and ln(t) are linear independent solutions of the corresponding homogenous equation
b. using variation of parameters find the particular solution to the non homogenous equation
c. express the solution to the non homogenous equation in terms of a. and b.
d.since y itself does not appear in the equation, the substitution w=y' can be used to reduce the equation to a linear 1st order equation. use this substitution to solve for w directly using a 1st order technique and verify that the two techniques produce the same answer
In: Advanced Math