Solutions of Linear Diophantine Equation ax + by = c

If (xo,yo) is a solution of the Linear Diophantine equation ax + by = c , then the set of solutions of the equation consists of all integer pairs (x,y), where either x = xo + tb/d and y = yo - ta/d ( t = ........,-2,-1,0,1,2,........)
or , x = xo - tb/d and y = yo + ta/d ( t = .........,-2,-1,0,1,2,.......) where d = g.c.d.(a,b).

Solution Summary

This solution is comprised of a detailed explanation for the Linear Diophantine Equation.
It contains step-by-step explanation for if (xo,yo) is a solution of the Linear Diophantine equation ax + by = c ,
then the set of solutions of the equation consists of all integer pairs (x,y), where
either x = xo + tb/d and y = yo - ta/d ( t = ........,-2,-1,0,1,2,........)
or , x = xo - tb/d and y = yo + ta/d ( t = .........,-2,-1,0,1,2,.......) where d = g.c.d.(a,b).

One fundamental difference between a linearequation and a linear inequality lies in the number of possible solutions. A linearequation has a finite (limited) number of solutions while the linear inequality has a range (set) of solutions.
I need an example of an equation and an inequality that expresses the above difference

See attached
1- Prove by induction, the following result for all n
1- Prove by induction, the following result for all n
2- Prove that , if x2
3- Use induction to prove that
(This can be used to show that the infinite har

The techniques for solving linearequations and linear inequalities are similar, yet different. Explain and give an example of both a linearequation and a linear inequality that demonstrates this difference.
1.) Solve and check the linearequation.
5x - 5 = 30
A) {30}
B) {34}
C) {11}
D) {7}
2.) Solve and check th

Consider first a linearequation of the form 2x - 5y = 8.
Now choose a linear inequality of the form 5y < 2x - 8 or 5y > 2x - 8.
What are the major differences between the linearequation graph and the linear inequality graph?

Please solve the following problems:
1. Compute the following ...
2. Let Fm be the set of all integral multiples of the integer m. Prove that ...
3. Draw the graphs of the straight lines defined by the following Diophantineequations ...
4. Prove that every integer is uniquely representable as the product of a non-negati

For each of the following ordinairy differential equations, indicate its order, whether it is linear or nonlinear, and whether it is autonomous or non-autonomous.
a) df/dx +f^2=0
(See attachment for all questions)

Consider the attached differential equation where I = (a,b) and p,q are continuous functions on I.
(a) Prove that if y1 and y2 both have a maximum at the same point in I, then they can not be a fundamental set of solutions for the attached equation.
(b) Let I = {see attachment}. Is {cos t, cos 2t} a fundamental set of solu