Question

How to find the general solution of a second order linear equation given three particular solutions?

Ly=f => This means it's a non-homogeneous right?

y=x , y=e^x+x , y=2e^x+x+1 are the three given particular solutions.

Answer 1

add them up ...
and also add the solution of the Ly=0 (homogeneous part)

Answer 2

how do you find the the solution of the homogeneous part?

Answer 3

y=yp+th => that's the form I would follow right?

Answer 4

you need to know what L is ...

Answer 5

*yp+yh

Answer 6

it's not given though...

Answer 7

the problem just say Ly=f

Answer 8

yeah ... sometimes it is also called complimentary function.

Answer 9

Here's the complete problem:

Three particular solutions of a certain second order linear equation Ly=f are y=x, y=e^x+x, y=2e^x+1+x. What is the general solution?

Answer 10

This problem from the section talking about the Wronskian

Answer 11

these particular solutions are due to superposition
y=x , y=e^x+x , y=2e^x+x+1

probably your question just expects to write
\[ c_f + y_p\]
where cf is complimentary function.

Answer 12

I don't exactly know what complimentary function means... Could you explain it to me/?

Answer 13

so my yp would = 3e^x+3x+1?

Answer 14

cf is just solution to homogeneous part.

Answer 15

the solution to Ly=0, correct? And how do I find that...?

Answer 16

I don't know how wronskian is related to this ... probably with variation of parameters. though if there.

Answer 17

so should my answer just be y=cf+3e^x+3x+1?

Answer 18

without L given , I don't think i can find out. Let me see if ... i can extract info out of variation of parameters.

Answer 19

ok, thank you.

Answer 20

man .. this is crazy ... can we find the linear operator with particular solution given
http://en.wikipedia.org/wiki/Variation_of_parameters

Answer 21

why do you have three particular solutions first of all?
y=x , y=e^x+x , y=2e^x+x+1

Answer 22

are these all particular solution of superposition of different inputs or .... two of them is part of solution of homogeneous equation?

Answer 23

We haven't talked about variations of parameters though...

Answer 24

Doesn't say anything about that. Those yp's were just given.

Answer 25

Three particular solutions of a certain second order linear equation Ly=f are y=x, y=e^x+x, y=2e^x+1+x. What is the general solution?

Answer 26

That's the complete problem.

Answer 27

and f is also not given!! ... probably your answer is
\[ y_h + y_1 + y_2 + y_3\]
what level do you study man??

Answer 28

College? I am currently looking into using wronskians with three functions. Maybe that would help me.

Answer 29

The book we're using is from 1960's!!

Answer 30

It doesn't give a lot of examples to work with before giving exercises.

Answer 31

yeah i know you don't encounter these until you second year in university ... extracting info out of solution ... i've not tried yet.

Answer 32

I don't know much about wronskian but i know it is used in method variation of parameters to find particular solution. also to check linear dependence between solutions.

Answer 33

Well, I really appreciate your help man. At least I know I just need to add up all those yps.

Answer 34

http://www.math.ucdavis.edu/~dragon/22B07/wronskian.pdf <= Looks like this would lead me to the right track

Answer 35

well ... as far as i know ...and thought,

I thought in this way.