undetermined coefficients approaches and variation of parameters

[PG1995]

Hi

Linear non-homogeneous differential equations are solved using the following two techniques in the book: (1) Undetermined Coefficients (superposition and annihilator approaches), (2) Variation of Parameters.

The book says undetermined coefficients approaches do not apply when the forcing function (e.g. g(x) in this equation ay'' + by' + cy = g(x)) is: In(x), 1/x, tan(x), arcsin(x) and so on. And in such cases variation of parameters technique is used.

Q1: Does this mean variation of parameters (VP) method is superior to undetermined coefficients (UC) method in that it can applied to cases where UC method fails?

Q2: Please have a look on the attachment. You can see the formula for solving linear non-homogeneous diff. eq. there. Can I apply that formula to the 2nd order equations with getting the same results as with methods UC and VP?

Q3: Is there an easy way to extend the formula in the attachment to apply to higher order equations (4th order is highest I'm interested in)?

Please help me with the queries above. Thank you very much for all the help.

Regards
PG

 

[Ratchit]

PG,

Q1: Does this mean variation of parameters (VP) method is superior to undetermined coefficients (UC) method in that it can applied to cases where UC method fails?

You can solve equations with the VP method that cannot be solved with the UC method. Therefore, the VP method is superior. However, you will usually find that the UC method involves less work if it can be used. So take your choice.

Q2: Please have a look on the attachment. You can see the formula for solving linear non-homogeneous diff. eq. there. Can I apply that formula to the 2nd order equations with getting the same results as with methods UC and VP?

Yes, linear equations of any order have only one solution for a particular set of initial conditions, no matter which method is used to find the solution. If you can calculate those integrals, you should be able to solve any second order linear differential with constant coefficients.

Q3: Is there an easy way to extend the formula in the attachment to apply to higher order equations (4th order is highest I'm interested in)?

Yes, everything depends on what R(x) is. You can find scads of those formulas in books like The Standard Mathematical Tables published by the Chemical Rubber Co. **broken link removed** . Check out the table of contents, the analysis section.

Ratch

 

[PG1995]

Hi Ratch

I had checked the book you mentioned before I posted here. I looked up 31st edition. The chapter #1 "Analysis" does not give any formulae about differential equations or calculus. In chapter #5 "Continuous Mathematics" it gives formulas on differential equations but it restrict those only to 2nd order non-homogeneous. What edition did you have in mind? Please let me know. Thank you.

Best regards
PG

 

[PG1995]

Please have a look on the attachment. Please help me with those queries. Thank you.

Best regards
PG

 

[PG1995]

Hi

I hope someone is not in process of reply to the queries in my last post. I have found the answers to the queries. Thanks for giving it a look.

Regards
PG

 

[Ratchit]

PG,

I had checked the book you mentioned before I posted here. I looked up 31st edition. The chapter #1 "Analysis" does not give any formulae about differential equations or calculus. In chapter #5 "Continuous Mathematics" it gives formulas on differential equations but it restrict those only to 2nd order non-homogeneous. What edition did you have in mind? Please let me know. Thank you.

I own the older 20th edition of that book. There is a section called "Differential Equations" that give all kinds of formulas for linear equations including those of higher orders. I don't know what the later editions of the book contain.

Ratch