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Define functions $f_1,f_2 : \mathbb{R} \rightarrow \mathbb{R}$ by asserting that for all $x \in \mathbb{R}$ it holds that: $$f_1(x) = e^{2x}\cos x, f_2(x) = e^{2x}\sin x.$$

Also, define a set $$F = \{f : \mathbb{R} \rightarrow \mathbb{R}\,|\,f''-4f'+5f=0\}.$$

It is easy to see that for all $i \in \{1,2\}$ it holds that $f_i \in F$. How can I show that $\{f_1,f_2\}$ is a basis for $F$?

Edit: Here's an incomplete attempt at a proof. The starred lines (namely, 2 and 5) are the ones I need help with.

  1. Let $F' = \{c_1 f_1 + c_2 f_2 \,|\,c_1,c_2 \in \mathbb{R}\}$.
  2. Prove that $\{f_1,f_2\}$ are linearly independent.*
  3. Conclude that $\{f_1,f_2\}$ is a basis for $F'$, and therefore that that $\mathrm{dim}(F')=2$.
  4. Prove that $F' \subseteq F$. (This is easy.)
  5. Use a theorem to show that $\mathrm{dim}(F)=2$.*
  6. Conclude that $F'=F$, and therefore that $\{f_1,f_2\}$ is a basis for $F$.

Line 5 is the really interesting one. What's the name of the theorem that tells us that $\mathrm{dim}(F) = 2$?

Edit 2: Line 5 follows from Theorem 3.4 in this document. (Thank you muzzlater.) So all that remains to show is Line 2.

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math.uconn.edu/~kconrad/blurbs/linmultialg/diffeqdim.pdf. This page goes into detail about the dimension of the solution space for a linear ODE. –  muzzlator Mar 4 '13 at 6:22
    
Didn't you forget "$=0$" in definition of your $F$? –  Kaster Mar 4 '13 at 6:40
    
@Kaster Thanks, i new something looked odd. –  goblin Mar 4 '13 at 6:41
    
@muzzlator Can I have a bit more help? The document you linked is pretty hard going, and I can't find the result I need... –  goblin Mar 4 '13 at 8:01
1  
@muzzlator I found it, its Theorem 3.4 in that document. –  goblin Mar 4 '13 at 8:07

1 Answer 1

Below is a proof of the uniqueness theorem, using Wronskians and variation of parameters.

Theorem $\ $ If $\rm\:f,g,h\: $ are solutions on an interval I of

$$\rm y'' =\ p\ y' + q\ y,\ \ \ \ p,q\ \ continuous\ on\ I $$

and the Wronskian $\rm\ \ W = g\:h'-g'h \ne 0\:$ for all $\rm\:x\in I$

then $\,\exists\,$ constants $\rm\: c,d\:$ such that $\rm\: f = c\: g + d\: h\:$ on $\rm\,I.$

Proof $\ $ The equations $[0],[1]$ below have unique solution $\rm\:(c,d)\:$ via det $\rm = W \ne 0\:$ on $\rm\,I.$

$\rm[0]\qquad f\ =\ c\: g \: + d\: h $

$\rm[1]\qquad f' =\ c\: g' + d\: h'$

Now $\rm\:q\:[0] + p\:[1]\ $ yields, $ $ on $ $ LHS: $\rm\,\ q\:f+p\:f'\: =\ f'',\ $ similar on RHS below

$\rm[2]\qquad f'' =\ c\: g'' + d\: h''\ $ via RHS: $\rm\ \, q\:g+p\:g'\: =\ g'',\,\ \ q\:h+p\:h'\: =\ h''$

$\rm[3]\qquad 0\ =\ c'\:g \:+ d'\:h\:\ \ $ via $\ \ [0]'-[1]$

$\rm[4]\qquad 0\ =\ c'\:g' + d'\:h'\ \ $ via $\ \ [1]'-[2]$

$[3],[4]\:$ have solution $\rm\:(c',d') = (0,0),\:$ which is unique by $\rm\ det = W = g\:h'-g'\:h \ne 0\:$ on $\rm\,I.\:$ Therefore $\rm\:c,d\:$ are constants. $\ \ $ QED

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Where was continuity of $\rm{p},\rm{q}$ used? –  goblin Mar 5 '13 at 1:45

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