Hw.1. Solve the integral equation ∫0^1 g(s) = f(s) + λ∫0^1...

Hw.1. Solve the integral equation $\int_0^1 g(s) = f(s) + \lambda \int_0^1 (s + t) g(t)dt$ Hw.2. Consider the following integral equation $g(s) = 1 + \lambda \int_0^1 (st) g(t)dt$ Make use of the relation $|\lambda| < B^{-1}$ To show that the iterative procedure is valid for $|\lambda| < 3$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Transcript

00:01 Hi there in this question we're given that r is the relation on n where x is related to y if and only if x is equal to k y for some k belong to the set of all natural numbers.

00:10 So here we have to check whether this relation is reflexive, symmetric and transitive.

00:15 So let's see how we'll do this.

00:20 So first we'll check whether the relation is reflexive for that.

00:24 First let's check what is reflexive relation.

00:26 A relation are on a set a is reflexive if a is related to a for all a belong to a.

00:33 So every element of a should be related to itself.

00:36 So for here we have a to be the set of all natural numbers actually.

00:41 So here we have n.

00:43 So let let x belong to n.

00:48 So we are taking an arbitrary element from the set of all natural numbers.

00:52 And now we are seeing whether it is related to itself.

00:55 So for that we can say we have x is equal to 1 multiplied by x.

01:03 So here for k equal to 1, we have the condition satisfied.

01:10 That is for k equal to 1, the condition is getting satisfied.

01:12 That is x is equal to k y for some n.

01:16 K belong to n.

01:17 Therefore, we have that this is x is related to x.

01:23 So since x was taken as arbitrarily, so r is reflexive.

01:28 So we can say that r is reflexive.

01:33 Now we'll see the definition of symmetric...

Question

Hw.1. Solve the integral equation $\int_0^1 g(s) = f(s) + \lambda \int_0^1 (s + t) g(t)dt$ Hw.2. Consider the following integral equation $g(s) = 1 + \lambda \int_0^1 (st) g(t)dt$ Make use of the relation $|\lambda| < B^{-1}$ To show that the iterative procedure is valid for $|\lambda| < 3$

Please give Ace some feedback