Let ϕ be the solution of the integral equation . Then
- ϕ(0)=20exp(-1) – 8
- ϕ(0)=22 – 8e
- ϕ(0)=22 – 8exp(-1)
Let . Substituting in the equation,
Concluding, ϕ(0)=20exp(-1) – 8.
For a Fredholm integral equation of 2nd Kind, using , where
Here, . Hence
Hence the solution is and you know how to finish the rest.
Using iterated kernels,
We will get ,
Here, and you know where to take it from here.
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!]
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).