Reference Request for Linear ODEs

Question

Homogeneous, linear ODEs of the form $$\mathrm{f}^{(n)}(x)+a_{n-1}\mathrm{f}^{(n-1)}(x)+\cdots+a_1\mathrm{f}'(x)+a_0\mathrm{f}(x)=0$$ where each $a_i \in \mathbb{R}$ are known to have "solution spaces" of dimension $n$.

For example, the first order ODE $$\mathrm{f}'(x)+\mathrm{f}(x)=0$$ has the solution space $\mathrm{f}(x)=k\,\mathrm{e}^{-x}$ which is spanned by $\mathrm{e}^{-x}$. The second order ODE $$\mathrm{f}''(x)+\mathrm{f}(x)=0$$ has the solution space $\mathrm{f}(x)=a\sin x + b\cos x$ which is spanned by $\sin x$ and $\cos x$.

What is the name of this result, and can someone please recommend a good on-line reference?

Answer 1

The solution space $\mathscr X$ of $$ \mathrm{f}^{(n)}(x)+a_{n-1}\mathrm{f}^{(n-1)}(x)+\cdots+a_1\mathrm{f}'(x)+a_0\mathrm{f}(x)=0, $$ is an $n-$dimensional subspace of $C^\infty(\mathbb R)$.

I can only think of rather advanced textbooks for reference. The one I prefer is the textbook of Hartmann.

The best way to show that $\mathscr X$ is $n-$dimensional is the following:

a. For every $\xi_1,\ldots,\xi_n$ the initial value problem $$ \mathrm{f}^{(n)}(x)+a_{n-1}\mathrm{f}^{(n-1)}(x)+\cdots+a_1\mathrm{f}'(x)+a_0\mathrm{f}(x)=0, \quad f^{(0)}=\xi_1,\ldots,f^{(n-1)}(0)=\xi_{n}, $$ possesses a unique, and global solution $\varphi=\varphi(t;\xi_1,\ldots,\xi_n)$, which is linear with respect to $\xi_1,\ldots,\xi_n$, and hence $$ \varphi(t;\xi_1,\ldots,\xi_n)=\sum_{k=1}^n\xi_k\varphi(t;\delta_{1k},\ldots,\delta_{1n}) =\sum_{k=1}\xi_k\varphi_k(t). $$ So, these $\varphi_1,\ldots,\varphi_n$ span $\mathscr X$. They are linearly dependent because if $$ c_1\varphi_1(t)+\cdots+c_n\varphi_n(t)=0, \quad \text{for all $t$}, $$ then $$ \varphi(t;c_1,\ldots,c_n)=c_1\varphi_1(t)+\cdots+c_n\varphi_n(t)=0, \quad \text{for all $t$}, $$ and thus for every $k=1,\ldots,n$ we would have $\varphi^{(k-1)}(0;c_1,\ldots,c_n)=0$. But $$ \varphi^{(k-1)}(0;c_1,\ldots,c_n)=c_k. $$

Answer 2

This is the uniqueness theorem for linear differential equations. It is usually proved by variation of parameters using Wronskians, as is done below The proof below easily generalizes to higher order. An analogous proof works also for difference equations (recurrences).

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

References

L. E. Pursell. A simple uniqueness theory for ordinary linear homogeneous differential equations, Amer. Math. Monthly, 74, 1967, 47-50

Variation of Parameters:
https://planetmath.org/variationofparameters http://ltcconline.net/greenl/courses/204/appsHigherOrder/variationHigher.htm

Marius van der Put. Symbolic analysis of differential equations.

This post: 2003-11-12, There are no other solutions... how to prove it?

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The above post is excerpted from my sci.math post on Apr 27 2004 in the thread "number of indep. soloutions to diffyqs?" (sic)