1. E

    Resources for questions on primitive roots

    Does anyone know of any resources on questions on primitive roots and order of a modulo n? They need to be suitable for elementary number theory course. (These could be interesting results and challenging ones).
  2. E

    What is the most motivating way to introduce primitive roots?

    I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact. These...
  3. E

    Normalization and Scaling of second order DE

    I understand how to normalize a second order system, but I wanted to know if the same steps are taken when the parameters of the system are not scalar but matrices. For example d2X/dt2 + (2H)dX/dt + (E)X = 0 where 2H and E are 3x3 matrices and X is a 3x1 column vector. The solution I have...
  4. S

    solve the 1st order unforced differential equation

    solve the 1st order homogeneous differential equation $2t{w}'-3w=0$ WORKED ANSWER For a 1st order homogeneous ODE, have solution formula: $A\cdot e^{-\int p}$ So we are doing the following integration $e^{-\int \frac{3}{2t}}\cdot A$ $\Rightarrow \int 3\cdot \frac{1}{t}\cdot...
  5. D

    solving an equation using order of operations

  6. M

    first order differential equation help..

    having problems with solving these differential equations... will be very thankful anybody can solve these or give an idea to solve y' =tan (x+y) -1 (x^4 -2xy^2 +y^4 )dx - (2x^2 y -4xy^3 )dy = 0 (y sec^2 x +secx tanx ) dx +(tanx +2y )dy =0
  7. A

    I need to re-learn all highschool level algebra in order to attend college.(help)

    I will just shoot straight: I am 30 years old. I have not done any type of true mathematics in over 10 years. In the preceding decade, I have forgotten everything I've ever known about mathematics. Do any of you know of any sites that offer free lessons in algebra 1, geometry, etc...
  8. H

    Need help with negative numbers and order of operation

    Ex. 3= -7 - (-x) Please help me with the steps. I think the first thing I should do is get "x" by itself. To get x by itself , the way my mind is telling me to do is is to add "-x" to both sides of the equation ,giving me 3x =-7 . I don't even know how to continue with this because I'm...
  9. M

    Second Order Differential

    In a galvanometer the deflection θ satisfies the following differential equation:- (d2θ/dt2) + 2(dθ/dt) + θ = 4 Solve the equation for θ given that when t=0, θ=0 and dθ/dt=0 Thanks in advance!
  10. B

    Second Order ODE second term tending to 0 - question reads something must be unsolve

    Hi, I have a 2nd order ODE of general form: y= Ae^(17x) + Be^(-x) Initial conditions are: y(0)=0; and y'(0)=2 From this I got A = 1/9 and B= -1/9 Leading to: 1/9e^(17x) - 1/9e^(-x) Even I can see the second term is tending to zero - as it was when I tested a few points. In the question...
  11. B

    2nd Order ODE's

    Hi, I have to solve: 1. y''+4y'+15y=0 Multiplying through used y=e^(landa x): Gives essentially the quadratic: y^2+4y+25=0 This gives two complex roots being from what I can see -2 plus or minus i/2sqrt 84. Solving the DE then gives y(x)=e^-1/2x(Acos(1/2sqrt84x+Bsin1/2sqrt84x) Does that...
  12. P

    Order For Transformations / Translations?

    Hello, I am struggling with doing transformations and translations of the graphs of functions. Specifically I don't understand the order I am supposed to do them in. I have had to do problems where I am given the graph of a function and then I am asked to select the correct new graph based on...
  13. E

    Two cyclic modules are isomorphic iff they have the same order ideal.

    Every cyclic R-module V=<v> is R-isomorphic to R/ord(v). Conclude that two cyclic modules are R-isomorphic if and only if they have generators with the same order ideal. Here ord(v) = {r in R: rv=0}. f: <v> --> R/ord(v), f(rv) = r + ord(v) is well defined and is an R-isomorphism. If ord(v) =...
  14. S

    Second Order homogeneous ODE

    Have reached this stage of an ODE problem x''(t)=2*sin(theta) * -2*sin(theta)*x(t) , where 2*sin(theta) is a constant Would x'(t) simply be: x'(t)= 2*sin(theta) * -2*sin(theta) *x(t) * t + C ?
  15. S

    2nd order ODE

    Given y''(t)+7x'(t)= 0 with x'(0)=a constant U Solve for y' Can l integrate the above ODE directly with respect to a variable t? hence, y'(t)=-7x(t) Thanks
  16. K

    How to convert to first order system?

    Hello! I have to convert this equation into a first order system and then write the discrete equations for that system based on the Euler Method as explicitly as possible. Then I have to apply one step of Euler's method by hand with h=0.1. I think I can do the Euler step by myself, but am...
  17. E

    If p and q are primes then every group of order p^m q^n is solvable.

    Hi: Burnside proved (using representation theory) that the number of elements in a conjugacy class of a finite simple group can never be a prime power larger than 1. Use this fact to prove Burnside's theorem: If p and q are primes then every group of order p^m q^n is solvable. What I did: let...
  18. X

    homogenous first order ordinary differential equation

    for the first photo , why when the G=0 , it is called homogenous system ? I was told that if a first is homogenous , the power of x and y terms must be the same ?
  19. S

    First Order Differential Equation

    I'm having a problem with this question, at the point I've got to I've ended up with ln(0) which isn't possible, working below: 1/t dx = x/1-t dt t dx = 1-t/x x dx = 1-t/t After integrating: x^2/2 = ln(t) - t + C Solving for x: x^2 = 2ln(t) - 2t + D x = sqrt(2ln(t) - 2t + D) Solving for...
  20. S

    First Order Differential Equation

    Hi, I don't understand the second blue line in this solution. Why is the C suddenly C' and why is it then Cx? Thanks!