1. Modified Bessel’s Equation Danh Tran 2012951269

2. Of Order n From proving Bessel equation, we know that the general solution is in the form of A and B are arbitrary constants J 0 is the Bessel Function of the First Kind of Order Zero. Y 0 is the Bessel Function of the Second Kind of Order Zero. General Form

3. Modified Version 1 LetSo that we can reach the the general form and general solution for that is known. So we need to put everything in terms of t (2) (3) (1)

4. Substituting (2) and (3) with After substituting, we have the general form and solution in terms of t Where andinto (1)

5. Modified Equation 2 Let Then (4) (5) (6)

6. Substituting Substituting (5) and (6) with andinto (4) Simplify by dividing

7. Let Then Dividing by which simplifies to

8. From this we can modify the generation solution Where Solution to equation of Modified Bessel Equation 2 is Where

9. Bessel Equation 3 Which can be written as Where Form (a)

10. After substituting, we have the general form and solution in terms of t Where Since in the solution exist a imaginary number we have to change it to a more preferred system The relation between J and Y and I and are linearly independent, we can convert C and D are arbitrary Constants

11. Bessel Equation 4 However it can be rewritten by multiplying x 2 Form (7)

12. Let (8) (9) Substituting (8) and (9) with andinto (7)

13. Where Therefore the solution is since In terms Because t is a factor of x The general solution for (10) is found by the process of Bessel Equation 3 (10) Where n=1