## Question 2

Let ϕ be the solution of the integral equation . Then

- ϕ(0)=20exp(-1) – 8
- ϕ(0)=20e-8
- ϕ(0)=22 – 8e
- ϕ(0)=22 – 8exp(-1)

### Solution 1

Let . Substituting in the equation,

Hence,

Concluding, ϕ(0)=20exp(-1) – 8.

### Solution 2

For a Fredholm integral equation of 2nd Kind, using , where

Here, . Hence

Hence the solution is and you know how to finish the rest.

### Solution 3

Using iterated kernels,

where

We will get ,

Here, and you know where to take it from here.

### Observation

Use Leibniz Integral Rule (link to blog page quoting the result) – Wikipedia (In particular, a specific case called Fundamental Theorem of Calculus here, since limits are constant) to find derivative of the equation w.r to x. What do you observe? *[Hint: you will get a differential equation, as – try solving it!] *

## Question 4

### Method

Solve the corresponding transposed homogenous integral equation, and find corresponding solution for .

Then, for each option, if . If it is zero, it has solutions, and if it is not zero, no solutions.

Here, solution, and hence options 2,3 and 4. ( and so on).