Question

Upon solving a differential equation... how can you show that the solution you found is the ONLY solution to the problem? Is it possible to prove?

Answer 1

finding a solution proves existence
proving uniqueness is a different story i think

Answer 2

i think this is covered in real analysis
@zzr0ck3r might be familiar with existence uniqueness thm proof

Answer 3

thank you so much! I had a feeling this might not be a possibility but I'd like to explore this to the end. Thanks again for direction :D

Answer 4

I have only had one undergraduate dif eq class :( sorry

Answer 5

But it does seem like a big question. I imagine it is different for all the different types.

Answer 6

Ok cool thanks! Even if I don't figure out exactly how to do it, I want to get a taste for how far it's possible to go on what is known so far so again, thank you for the suggestions!

For some context the reason I'm interested is because diffeqs solve the schrodinger equations as you probably know... and I'd like to get a better feel of the solutions and what they look like, and why they are the way they are.

Answer 7

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB4QFjAA&url=http%3A%2F%2Fwww.math.uconn.edu%2F~troby%2FMath200S08%2Fexistence.pdf&ei=IbeKVJvRCNfgoATiiIKoAQ&usg=AFQjCNGulI36dNgDogryLCe0NpMAkXREZQ&bvm=bv.81828268,d.cGU

paper on methods of first order

Answer 8

Ahh in that case I suggest you look at the phase line graph thingys

Answer 9

Phase line graph... you don't say? *opens new tab

Answer 10

I forget the name, the one with all the little arrows

Answer 11

Direction Fields

Answer 12

I can't quite figure out how the graphs will help me yet but I gotta keep them in mind too. I will check out the paper in the meantime.

Answer 13

@diracdelta Please note that if your differential equation is a problem of Cauchy, namely you also have initial conditions, then the solution you will get is unique, by a theorem of Mathematical analysis

Answer 14

Yes I definitely have initial conditions, so there's a theorem that can prove that my solution is the only possible one? (And by my solution I don't ACTUALLY mean mine haha...)

Answer 15

listen to me

Answer 16

i think i need to explain the diracdelta function to you okay

Answer 17

you see its simple, convolving with 1 is same thing as conving with the irac delta function

Answer 18

u'(t)=dirac(t)

Answer 19

u(t) is the unit step function
u=0,u<0
u=1,u>=0

Answer 20

or u=0,u<=0
u=1,u>0