course mth174
Hopefully, I can take test 3 tomorrow. Here are the few things I have been having trouble understanding so far:I know how to use Euler's method for finding the approximate solution of differential equations, but this is kind of confusing me.
when you give a fuction dy/dx=3.3x/ e^ y^2 on the interval (.75,1.35) if (.75)= 1.25 using increment .15
My understanding is that you find the slope, multiply by the increment and this is your new d'y value,then you add that to the y coordinate and then add your increment to the x coordinate and you have your new coordinates. You follow this process until you reach the point. But here, you're already starting at .75 and it already gives the .75 value, so how would you continue on out if you already having the information needed to reach it. Or would you start on these intervals and continue on out until you reach the y value of 1.25?
Sounds like you've got it; I'm not sure what you're missing, but here are some details, which will hopefully fill in anything you're missing: Start with x = .75 and y = 1.25, plug into the formula to find dy/dx, follow that slope for `dx = .15 and get the approximate value for x = .75 + `dx .75 + .15 = .90. Then use the new y value, with x = .90, and repeat the process. Keep going until you reach x = 1.35.
When constructing a fourier series by the coefficients, I've read the chapter *hopefully* to have a thorough understanding of what they mean and how they are used. But when constructing the graphs, do we need to work backwards?
I know that a(O)= 1/2pi integral of f(x) dx from -pi to pi
an is 1/pi integral of f(x) cos(nx) from "" ""
and the same for bn except its f(x)sin(nx) from """"
(Now im understanding the concept of what we went over (ie. why the value of the integral of cosnxsimx is zero from -pi to pi
Given a(0)=1
an(pi/ sqrt(n) and b(n)= 0
will my fourier series just consit of a(O) + integral(f(x) cosnx) from -pi to pi ?
For each value of n you have to perform the integral. 1 / pi multiplied by the integral gives you the coefficient for n. Then you construct the Fourier series for these coefficients.
Hopefully this will become clear tonight as I review.
For the interval of convergence of x/ .8^2.5 + x^2/2.4 ^2.5 + x^3 / 7.2^ 2.5
pattern is 1/3 of the preceding amount
a(n) = 1/ .8 ^2.5 * 3^(n-1)
a(n)= [1/ .8^2.5 * 3^(n-1+1)] / [1/ .8^2.5* 3^(n-1)
Right, but isn't this last one a(n+1)?
simplifying gives 3^n-1 / 3^n =1/3
radius of convergence 1/ 1/3= 3
power series in form of a(n) * (x-0) ^n
center of interval convergence is x=0
its from -3< x< 3
Another note:
I'm incorrectly integrating. When I'm doing air resistance and get it down to the form of dv/ (g- k/m v^2) , would I have to use a trigonometric identity? let v=sinx then solve?
The denominator is the difference of two squares. Factor into (sqrt(g) + sqrt(k/m) v) ( sqrt(g) - sqrt(k/m) v), then use partial fractions.
If you prefer a trig substitution factor out the g to get g ( 1 - k / (mg) v^2) then let v = sqrt(mg / k) sin(x), in order to get 1 - sin^2(x) in the denominator.
The problem concerning the voltage across a circuit, the more I look over the more I understand but how is this related to modeling it off of frequency? If we are given certain conditions and asked to write a differential equations modeling that, what is the relationship?
Not sure what you mean by 'modeling it off of frequency'. There are situations where you have a driving voltage which oscillates at a given frequency, but those equations are beyond the scope of this introduction (they occur in a standard first course in ordinary differential equations).