## Find the set of solutions of each of the linear co...

Find the set of solutions of each of the linear congruence:

a) x≡3x≡3 (mod 5).

b) 2x≡52x≡5 (mod 9).

## How to solve Laplace transformations of systems of...

My question is: Use the laplace transform to solve the system.

dx/dt + d^2y/dt^2 = 5e^(2t)

dx/dt - x - dy/dt + y = 8e^(2t)

x(0) = 2, y(0) = 1, y'(0) = 1

What I've done in Maple:

with(inttrans);
with(DEtools);
eq5 := (diff(x(t), t)+diff(y(t), t\$2) = 5*exp(2*t), t, s);

eq5s := laplace(%, t, s);

eq6 := (diff(x(t), t)-x-(diff(y(t), t))+y = 8*exp(2*t), t, s);

eq6s := laplace(%, t, s);

solve({eq5s, eq6s}, {laplace(x(t), t, s), laplace(y(t), t, s)});

subs({x(0) = 2, y(0) = 1, (D(y))(0) = 1}, %);

eq3 := invlaplace(%, s, t);

How do I simplify?  If you plug it into maple I come up with an answer that has x and y on each side.  I guess I'm just wondering how I can set them equal to each other to solve and get rid of the variable x and y.  I know answer is correct as I've also ran it through ODEtest.  Please help.

However you figure out getting rid of the variables I assume will help me also in solving the next problem:

Use the Laplace Transform to solve the system

dx/dt = 7x - y + 6z

dy/dt = -10x + 4y - 12z

dz/dt = -2x + y - z

x(0) = 5, y(0) = 7, z(0) = 2

I have attempted the second problem much like the first.  Thank you for your time.

## how to create a matrices by eigenvalues...

Generate 8 random 3 by 3 matrices using the RandomMatrix command from the  LinearAlgebra package. As each matrix is generated use Eigenvalues to compute its eigenvalues. Then take the product of the eigenvalues, and check that for each matrix, this product is equal to the determinant of the matrix.

## Problem with Paraboloid...

How can I plot a paraboloid?

## Analytical Solution...

This question explores the family of differential equations dy/dx=sqrt(􏰐 1 +􏰏( a*x )+ 􏰏 (2 *y)) for various values of the parameter a.

For the case a = 􏰐 0 find the analytical solution that passes through the point (0, 1) and verify that this is a solution to the differential equation. Use this solution to find the value of y correct to 4 decimal placeswhen x=􏰐1.

In maple i did

y:=(1/2)*x^2+sqrt(3)*x+1:
diff(y,x)

i got the answer x + sqrt(3)

as shown in the markscheme. please cluld anyone help how to get y before this step and what to do after.

## Crypt and decrypt with Vigenère...

```I have to crypt and decrypt with vigenere.
```

(procedures need lists)

```"In a Caesar cipher, each letter of the alphabet is shifted along some number of places; for example, in a Caesar cipher of shift 3, A would become D, B would become E, Y would become B and so on. The Vigenère cipher consists of several Caesar ciphers in sequence with different shift values.

To encrypt, a table of alphabets can be used, termed a tabula recta, Vigenère square, or Vigenère table. It consists of the alphabet written out 26 times in different rows, each alphabet shifted cyclically to the left compared to the previous alphabet, corresponding to the 26 possible Caesar ciphers. At different points in the encryption process, the cipher uses a different alphabet from one of the rows. The alphabet used at each point depends on a repeating keyword.[citation needed]

For example, suppose that the plaintext to be encrypted is:

ATTACKATDAWN
The person sending the message chooses a keyword and repeats it until it matches the length of the plaintext, for example, the keyword "LEMON":

LEMON
Each row starts with a key letter. The remainder of the row holds the letters A to Z (in shifted order). Although there are 26 key rows shown, you will only use as many keys (different alphabets) as there are unique letters in the key string, here just 5 keys, {L, E, M, O, N}. For successive letters of the message, we are going to take successive letters of the key string, and encipher each message letter using its corresponding key row. Choose the next letter of the key, go along that row to find the column heading that matches the message character; the letter at the intersection of [key-row, msg-col] is the enciphered letter.

For example, the first letter of the plaintext, A, is paired with L, the first letter of the key. So use row L and column A of the Vigenère square, namely L. Similarly, for the second letter of the plaintext, the second letter of the key is used; the letter at row E and column T is X. The rest of the plaintext is enciphered in a similar fashion:

Plaintext:	ATTACKATDAWN
Key:	LEMON
Ciphertext:	LXFOPVEFRNHR
Decryption is performed by going to the row in the table corresponding to the key, finding the position of the ciphertext letter in this row, and then using the column's label as the plaintext. For example, in row L (from LEMON), the ciphertext L appears in column A, which is the first plaintext letter. Next we go to row E (from LEMON), locate the ciphertext X which is found in column T, thus T is the second plaintext letter."```

I think that it can be done with a for loop but I do not know where to start.

## ELgamal procedure...

Hi I was wondering if anyone can help me with the following procedure, i am trying to write a procedure that can encrypt/decrypt messages encrypted using the method of ELGamal my current procedure runs but it takes too long to compute, and it can only decrypt.

My procedure is as follows:

my procedure for part c is as follows it seems to run but it takes a long time to carry out the procedure when i try to decrypt.

Elgamal := proc (ciphy, hkt, p, a, b)
local i, icdarray, s, q;
icdarray := Array(5 .. 388);
for i from 5 to 388 do
s := ciphy[i];
q := `mod`(1/hkt^proc3(a, b, p), p);
icdarray[i] := s*q;
end do;
return convert(icdarray, bytes);
end proc;

where proc3 is as follows

proc3 := proc (alpha, beta, p)
local k, R, i, j, N, A, t;
Description "baby step giant step procedure";
N := floor(sqrt(p-1))+1;
A := Array(0 .. N);
for j from 0 to N do
A[j] := `mod`(alpha&^j, p);
end do;
for i from 0 to N do
t := `mod`(beta*alpha&^(-N*i), p);
for k from 0 to N do
if t = A[k]
then return k+N*i;
end if;
end do;
end do;
end proc;

ciphertext:
[12432485341, 2579085006, 13736574369, 4105371047, 9573017222,

7824534168, 10017411248, 13292180343, 2356887993, 9573017222,

10017411248, 13765667419, 9795214235, 10017411248, 2801282019,

608404939, 4105371047, 13765667419, 11572790339, 13765667419,

11765894302, 10017411248, 13765667419, 4549765073, 10017411248,

13736574369, 2579085006, 4549765073, 10017411248, 4549765073,

13765667419, 2801282019, 830601952, 4105371047, 10017411248,

7824534168, 13765667419, 13736574369, 2801282019, 7824534168,

10017411248, 830601952, 9573017222, 4327568060, 13765667419,

6076051114, 8268928194, 13292180343, 10017411248, 7824534168,

386207926, 2801282019, 4105371047, 2579085006, 6076051114,

608404939, 13765667419, 6076051114, 830601952, 13765667419,

4105371047, 11765894302, 10017411248, 13765667419, 13292180343,

13736574369, 10017411248, 608404939, 10017411248, 7824534168,

2134690980, 13765667419, 4105371047, 11765894302, 2801282019,

4105371047, 13765667419, 2579085006, 608404939, 13292180343,

11543697289, 2579085006, 7824534168, 10017411248, 4549765073,

13765667419, 4994159099, 5853854101, 6076051114, 830601952,

4327568060, 6076051114, 5853854101, 10017411248, 7824534168,

13765667419, 4105371047, 6076051114, 13765667419, 9573017222,

13292180343, 10017411248, 13765667419, 4105371047, 11765894302,

10017411248, 13765667419, 5853854101, 6076051114, 7824534168,

4549765073, 13765667419, 11572790339, 13765667419, 4105371047,

11765894302, 2801282019, 4105371047, 13765667419, 4105371047,

11765894302, 10017411248, 13765667419, 4327568060, 2801282019,

608404939, 4549765073, 13292180343, 13736574369, 2801282019,

11543697289, 10017411248, 13765667419, 5853854101, 2801282019,

13292180343, 13765667419, 11765894302, 6076051114, 7824534168,

7824534168, 2579085006, 8268928194, 4327568060, 2134690980,

13765667419, 11543697289, 7824534168, 10017411248, 13736574369,

2579085006, 11543697289, 2579085006, 4105371047, 6076051114,

9573017222, 13292180343, 2385981043, 13765667419, 3245676045,

9573017222, 2801282019, 2579085006, 608404939, 4105371047,

6105144164, 13765667419, 5853854101, 11765894302, 10017411248,

608404939, 13765667419, 9573017222, 13292180343, 10017411248,

4549765073, 13765667419, 4105371047, 6076051114, 13765667419,

4549765073, 10017411248, 13292180343, 13736574369, 7824534168,

2579085006, 8268928194, 10017411248, 13765667419, 4105371047,

11765894302, 10017411248, 13765667419, 6076051114, 13736574369,

13736574369, 2801282019, 13292180343, 2579085006, 6076051114,

608404939, 2801282019, 4327568060, 13765667419, 386207926,

2579085006, 4327568060, 4327568060, 2801282019, 6298248127,

10017411248, 13765667419, 4105371047, 11765894302, 7824534168,

6076051114, 9573017222, 6298248127, 11765894302, 13765667419,

5853854101, 11765894302, 2579085006, 13736574369, 11765894302,

13765667419, 4105371047, 11765894302, 10017411248, 2134690980,

13765667419, 11543697289, 2801282019, 13292180343, 13292180343,

10017411248, 4549765073, 6105144164, 13765667419, 9795214235,

10017411248, 2801282019, 608404939, 4105371047, 13765667419,

830601952, 10017411248, 386207926, 10017411248, 7824534168,

11572790339, 7824534168, 2579085006, 4549765073, 4549765073,

10017411248, 608404939, 13765667419, 2801282019, 608404939,

4549765073, 13765667419, 4105371047, 9573017222, 9795214235,

8268928194, 4327568060, 10017411248, 4549765073, 6076051114,

5853854101, 608404939, 2385981043, 13765667419, 4994159099,

5853854101, 6076051114, 830601952, 4327568060, 6076051114,

5853854101, 10017411248, 7824534168, 13765667419, 5853854101,

2801282019, 13292180343, 13765667419, 2801282019, 13765667419,

4105371047, 6076051114, 9573017222, 7824534168, 2579085006,

13292180343, 4105371047, 6105144164, 13765667419, 4105371047,

11765894302, 10017411248, 13765667419, 830601952, 2579085006,

7824534168, 13292180343, 4105371047, 13765667419, 10017411248,

386207926, 10017411248, 7824534168, 13765667419, 13292180343,

10017411248, 10017411248, 608404939, 13765667419, 6076051114,

608404939, 13765667419, 4105371047, 11765894302, 10017411248,

13765667419, 5438553125, 2579085006, 13292180343, 13736574369,

5853854101, 6076051114, 7824534168, 4327568060, 4549765073,

2385981043, 13765667419, 4994159099, 6076051114, 9573017222,

7824534168, 2579085006, 13292180343, 4105371047, 6105144164,

13765667419, 8713322220, 2579085006, 608404939, 13736574369,

10017411248, 5853854101, 2579085006, 608404939, 4549765073,

13765667419, 11765894302, 2801282019, 4549765073, 13765667419,

4549765073, 10017411248, 13736574369, 2579085006, 4549765073,

10017411248, 4549765073, 6105144164, 13765667419, 9795214235,

10017411248, 2801282019, 608404939, 4105371047, 13765667419,

8075824231, 2579085006, 4549765073, 2579085006, 6076051114,

4105371047, 8075824231, 2385981043]

[1]: https://i.stack.imgur.com/xY3zd.png
[2]: https://i.stack.imgur.com/0eYFM.png
[3]: https://i.stack.imgur.com/PMk7s.png

## Runge-Kutta in maple...

dy/dx=sqrt(1+(a*x)+(2*y))

for the case a=1, y=1 and x=0 construct a program for the runge-kutta method of order 2 with formulae as follows where f(x,y)=dy/dx.

k_1=h*f(x_n,y_n)

k_2=h*f(x_n+h,y_n+k_1)

y_(n+1)=y_n+1/2(k_1+k_2).

After creating a program obtain value of y correct to 4 decimal places when x=1 for h=0.1 and h =0.05.

## Component with cycle for...

Meanwhile, thank you so much for everything.
I know I'm asking a lot but if you have time, you can help me do this?

Building a system of interactive components that, taken a function, two points 'a' and 'b' values ​​and an integer n, the calculations point between a and b in which the function assumes the minimum value by using the following procedure:

• It divides the values ​​between a and b into n equal parts (these will distance the one with the other (b-n)/2);
• calculates the function in each of these points;
• located between these values,  what is the minimum (in case of a tie, take the one closest to a)

I think i have to create a vector for each  part and  prehaps with a fcycle for, calculate the function, finelly i'll use minimize with all function.
Do you think is the correct procedure? If yes, how can I do it?

## How to show this Q&A on Maple...

How can i answer iv on Maple?

## Tangent to a curve in maple...

A family of curves has polar equation r=cos^n (theta/n), 0<=theta,n*pi, where n is a positive even integer.

Previously Using t = theta as the parameter and finding  a parametric form of the equation of the family of curves it was shown that

dy/dx = (sin(t)sin(t/n)-cos(t)cos(t/n)) /( sin(t)cos(t/n)+cos(t)sin(t/n)).

Is it possible to show on Maple with a program that there are n+1 points where the tangent to the curve is paralell to the y axis?

## Prove dy/dx with parametric form of equation....

A family of curves has polar equation r=cos^n (theta/n), 0<=theta,n*pi, where n is a positive even integer.

Using t = theta as the parameter, find a parametric form of the equation of the family of curves and show that

dy/dx = (sin(t)sin(t/n)-cos(t)cos(t/n)) /( sin(t)cos(t/n)+cos(t)sin(t/n))

on maple i tried

x:=((cos(t/n))^n)*cos(t):

y:=((cos(t/n))^n)*sin(t):

w:=diff(x,t)

z:=diff(y,t)

z/w

and i never got the above answer so i did

simplify(z/w)

(cos(t/n)*sin(t/n)-sin(t)*cos(t))/(cos(t/n)^2-cos(t)^2)

## Program for algebraic equation with mod in maple...

I need help to create a program that will find all the positive integers n, where n < 1000, such that
(n 􏰀-1)!= 􏰁 􏰀-1 (mod n^2 ) . program has to be in full and state the values of n obtained.