order

1. 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. 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. 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. 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. solving an equation using order of operations

(2+1){[(−4+4)2+3](−1)−7}
6. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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!