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.
For the linear IE , is
Solution 1: Using Method 2
Solution 2: Using Method 4
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 .
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.
These questions are simply about finding the integral and rearranging it.
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.
Let be the solution of the integral equation . Then equals,
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.