Where can I find proofs for d. eq solutions?

[mike_]

Hello. Where can I find proofs for the solution of d. equations? I can find the solutions but I cannot find the proofs in any textbook.

Specifically, how can I prove the solution for:

y''+ay=0
y''+ay'+by=0

Thank you.

 

[rock.freak667]

if a and b are constants then you have a DE with constant coefficients and you can easily derive the solutions knowing that y=e^rx is a trial solution.

 

[HallsofIvy]

Any good (not introductory) text on differential equations will contain proofs. For example, it can be proved that the set of any nth order, linear, homogeneous equations forms a vector space of dimension n. That, in turn, implies that if you can find n independent solutions, the general solution will be a linear combination of those.

 

[epenguin]

Hello. Where can I find proofs for the solution of d. equations? I can find the solutions but I cannot find the proofs in any textbook.

Specifically, how can I prove the solution for:

y''+ay=0
y''+ay'+by=0

Thank you.

You can do the difficult bit and you ask us how to do the easy one?

If you know a solution, then if you know how to differentiate you should usually easily be able to prove it is a solution.

(I may be unpopular

for revealing that for many students that is all they need. And that most "finding" solutions is semi-systematic ways of recognising that you do already know them at least in outline and just need to hammer constants etc, to fit properly.)

Proofs you only need for more general statements like HOI's.

 

[mike_]

Thanks for the replys. What I am looking for is how to find the solution without knowing it in the first place. A and B are constants.

rock.freak667 mentioned y=e^rx which also requires euler's formula to solve the equations. I don't understand this method and would like an easy to understand explanation. Thanks.

Also can anyone recommend a book that has this solution discovered step by step? Thanks.

 

[disregardthat]

You can do the difficult bit and you ask us how to do the easy one?

If you know a solution, then if you know how to differentiate you should usually easily be able to prove it is a solution.

(I may be unpopular

for revealing that for many students that is all they need. And that most "finding" solutions is semi-systematic ways of recognising that you do already know them at least in outline and just need to hammer constants etc, to fit properly.)

Proofs you only need for more general statements like HOI's.

That depends whether you want to find the solution or a solution. You still depend on the uniqueness of your solution. Some semi-systematic methods may not yield all solutions. Knowing they form a vector space, then finding enough linearly independent solutions will suffice.

 

[epenguin]

That depends whether you want to find the solution or a solution. You still depend on the uniqueness of your solution. Some semi-systematic methods may not yield all solutions. Knowing they form a vector space, then finding enough linearly independent solutions will suffice.

OK true enough. Though we did manage to do the traditional lde's for scientists at uni with never a mention of vector spaces. Not ideal I guess.

The question seems to have changed - the OP started by saying he can get the solutions but not prove them, but later that seemed to change a bit.

I am saying for a section of say biology etc. students who might well run up against d.e.s just now and then, maybe not often but occasionally necessary, and struggle to see how that one could be solved, or follow the concepts of the solution method offered, that all they need to do is to check that it is a solution which they can do and that will get them through. Then I'll leave the didactic soundness of that to Profs and planners - but I think a lot of biology/biochemistry/chemistry teachers may tell you it's that or nothing.

 

[mike_]

Is trial and error the only way to discover the solution y=e^rx?

Are the equations separable?