## Question 1

**Question**: Let be the solution of the integral equation . Then

A couple of solutions to this problem are discussed in Unit Quiz 1 (Question 2, only difference is that options 3 and 4 are about value of phi at x=1), even using methods mentioned in Unit 2.

## Question 2

For the linear IE , is

- 1/2
- 2
- 3/2
- 4

#### Solution 1: Using Method 2

*Method description can be found here: Unit 2 Summary*

.

Similarly,

Thus,

and hence,

#### Solution 2: Using Method 4

*Method description can be found here: Unit 2 Summary*.

and .

Hence,

#### Solution 3: Taking constant c

Given . We need to find ; where resolvent kernel is nothing but a function such that

All given answers are constants, hence, substituting in above equation, we get . Now substituting this expression of in the original equation and simplifying, we get .

## Question 3

The integral equation for has the solution

#### Solution 1: Using Method 1

*Method description can be found here: Unit 1 Summary*.

Put . We get . Substituting it back in original equation,

Simplifying further, we get . Hence, , option 4.

#### Solution 2: Using Method 2

*Method description can be found here: Unit 2 Summary*.

Continuing, . Hence,

.

Hence, the solution, , option 4.

#### Solution 3: Using Method 4

*Method description can be found here: Unit 2 Summary*. *I leave it to reader to try solve using this method, ample enough samples are given above.*

#### Solution 4: Elimination method

Substitute each option, see which one satisfies the equation.

- , and it leads to , which is absurd.
- Note that options 2 and 3 are the same – . Substituting in the equation, where c is a constant – which again, doesn’t satisfy the equation.

Hence we have eliminated three options – since at least one option must be true, we are left with option 4.

## Questions 4,5

These questions are simply about finding the integral and rearranging it.

## Question 7

For the integral equation where , the iterated kernel is

This question should be solved using Method 2. *Method description can be found here: Unit 2 Summary*. I leave it to the reader to find the answer.

## Question 9

Let be the solution of the integral equation . Then equals,

- -1
- 0
- 1
- 2

Attempt to solve using other methods are left to the reader. Note that this is a Volterra IE.

#### Solution by Leibniz Integral Rule

Leibniz Integral Rule (link) talks about differentiation of integral. Differentiating this equation using the above rule w.r. to x, we get

, using the original equation, this reduces to . Trivially, .

If you are interested, please explore solution by Laplace transforms and we can further expand this article; useful for many others around the globe.