7. Let m be a fixed positive integer.
(a) Prove that no two among the integers 0, 1, 2, . . . , m − 1
are congruent to each other modulo m.
(b) Prove that every integer is congruent modulo m to one of 0,
1, 2, . . . , m − 1.
1. Consider the initial value problem y′ =1+y/t, y(1)=3
for1≤t≤2.
• Show that y(t) = t ln t + 3t is the solution to the initial
value problem.
• Write a program that implements Euler’s method and the 4th
order Runke-Kutta method for the above initial value problem. Use
your program to solve with h = 0.1 for Euler’s and h = 0.2 for
R-K.
• Include a printout of your code and a printout of the results
at each...
10. Solve the following initial value problem:
y''' − 2y '' + y ' = 2e ^x − 4e^ −x
y(0) = 3, y' (0) = 1, y''(0) = 6
BOTH LINES ARE PART OF A SYSTEM OF EQUATIONS
Consider the initial value problem
y′ = 18x − 3y, y(0) = 2
(a) Solve it as a linear 1st order ODE with the method of the
integrating factor.
(b) Solve it using a substitution method.
(c) Solve it using the Laplace transform.
Use the Laplace transform to solve the following initial value
problem,
y′′ − y′ − 30y = δ(t − 7),y(0) = 0, y′(0) = 0.
The solution is of the form ?[g(t)] h(t).
(a) Enter the function g(t) into the answer box below.
(b) Enter the function h(t) into the answer box below.